Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×#2215 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,17)(4,25)(5,12)(6,33)(7,37)(8,14)(9,16)(10,21)(11,28)(13,18)(15,34)(19,64)(20,54)(22,75)(23,56)(24,58)(26,46)(27,82)(29,86)(30,50)(31,88)(32,52)(35,90)(36,91)(38,51)(39,92)(40,70)(41,73)(42,97)(43,74)(44,99)(45,87)(47,61)(48,62)(49,63)(53,67)(55,59)(57,72)(60,78)(65,105)(66,77)(68,120)(69,106)(71,108)(76,107)(79,83)(80,122)(81,102)(84,89)(85,123)(93,125)(94,116)(95,119)(96,111)(98,110)(100,112)(101,113)(103,124)(104,121)(109,114)(115,118)(117,127)(126,128);(1,3,18,59,7,22,31,70,5,20,61,110,32,71,36,74,9,24,63,111,38,76,35,73,8,23,62,97,34,72,6,21)(2,10,33,57,15,42,48,56,14,41,90,107,51,96,49,58,16,43,91,108,52,98,47,54,12,40,88,75,37,55,13,17)(4,19,60,109,29,68,80,116,26,65,112,128,81,117,85,119,30,69,113,125,87,121,84,92,28,67,99,124,83,118,27,66)(11,39,89,104,45,93,101,106,50,95,123,127,102,126,100,105,46,94,122,120,86,114,78,64,25,77,82,115,79,103,44,53);(1,4)(2,11)(3,19)(5,26)(6,27)(7,29)(8,28)(9,30)(10,39)(12,25)(13,44)(14,50)(15,45)(16,46)(17,53)(18,60)(20,65)(21,66)(22,68)(23,67)(24,69)(31,80)(32,81)(33,89)(34,83)(35,84)(36,85)(37,79)(38,87)(40,77)(41,95)(42,93)(43,94)(47,78)(48,101)(49,100)(51,102)(52,86)(54,64)(55,103)(56,106)(57,104)(58,105)(59,109)(61,112)(62,99)(63,113)(70,116)(71,117)(72,118)(73,92)(74,119)(75,115)(76,121)(82,88)(90,123)(91,122)(96,126)(97,124)(98,114)(107,127)(108,120)(110,128)(111,125);(1,5,9,8)(2,12,16,14)(3,20,24,23)(4,26,30,28)(6,31,36,35)(7,32,38,34)(10,40,43,41)(11,25,46,50)(13,47,49,48)(15,37,52,51)(17,54,58,56)(18,61,63,62)(19,65,69,67)(21,70,74,73)(22,71,76,72)(27,80,85,84)(29,81,87,83)(33,88,91,90)(39,77,94,95)(42,55,98,96)(44,78,100,101)(45,79,86,102)(53,64,105,106)(57,75,108,107)(59,110,111,97)(60,112,113,99)(66,116,119,92)(68,117,121,118)(82,122,123,89)(93,103,114,126)(104,115,120,127)(109,128,125,124)Polycyclic presentation:
2^9|b^2=eg,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gh,h^a=hi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320566, #72320861, #72320882, … |
| maximal normal subgroup of | #72320566, #72320861, #72320882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420147);