Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2215 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12834 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 128 points:
(1,2)(3,16)(4,24,27,46)(5,29)(6,13)(7,36)(8,38)(9,15)(10,19)(11,26,44,34)(12,17)(14,30)(18,60,63,105)(20,52)(21,71)(22,73)(23,54)(25,77,80,123)(28,45,81,101)(31,83)(32,87)(33,85)(35,49)(37,90)(39,92,93,117)(40,66)(41,96)(42,68)(43,98,99,112)(47,57)(48,59)(50,62,103,69)(51,55)(53,65)(56,75,111,100)(58,84)(61,74,115,95)(64,104,116,127)(67,91)(70,107)(72,120)(76,88,89,78)(79,122,86,82)(94,126,125,108)(97,109)(102,124,121,128)(106,110)(113,118,119,114);(1,3,17,55,7,21,31,66,6,20,57,109,35,70,33,68,9,23,59,110,37,72,32,67,8,22,58,96,30,65,5,19)(2,10,29,53,14,41,84,73,38,91,87,120,90,106,48,54,15,42,85,107,49,97,47,52,13,40,83,71,36,51,12,16)(4,18,56,108,28,64,86,117,34,69,112,128,88,118,80,115,27,63,111,126,81,116,79,92,26,62,98,124,78,114,25,61)(11,39,82,104,45,94,75,60,24,74,77,113,76,121,99,103,44,93,122,127,101,125,100,105,46,95,123,119,89,102,43,50);(1,4)(2,11)(3,18)(5,25)(6,34)(7,28)(8,26)(9,27)(10,39)(12,43)(13,46)(14,45)(15,44)(16,50)(17,56)(19,61)(20,69)(21,64)(22,62)(23,63)(24,38)(29,82)(30,78)(31,86)(32,79)(33,80)(35,88)(36,89)(37,81)(40,95)(41,94)(42,93)(47,100)(48,99)(49,101)(51,102)(52,105)(53,104)(54,103)(55,108)(57,112)(58,98)(59,111)(60,73)(65,114)(66,117)(67,92)(68,115)(70,118)(71,119)(72,116)(74,91)(75,84)(76,90)(77,87)(83,123)(85,122)(96,124)(97,125)(106,121)(107,127)(109,128)(110,126)(113,120)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320566, #72320831, #72320882, … |
| maximal normal subgroup of | #72320566, #72320831, #72320882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59492);