Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2158 |
| 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 ◁ C2×M16 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 228 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:
(1,2,6,14,9,13,8,38)(3,16,20,55,23,54,22,73)(4,24,26,47,28,11,34,45)(5,29,31,85,33,84,32,88)(7,36,35,51,37,91,30,15)(10,39,42,68,41,67,92,19)(12,48,50,60,49,59,86,17)(18,61,63,105,65,52,69,103)(21,71,70,107,72,122,66,56)(25,77,79,124,80,83,87,125)(27,81,78,76,82,90,89,46)(40,93,95,62,74,116,97,117)(43,57,53,94,98,110,106,109)(44,99,100,58,75,111,101,112)(64,118,115,114,119,121,120,104)(96,108,113,127,123,128,102,126);(1,3,17,57,7,21,31,39,6,20,48,94,35,70,33,68,9,23,60,110,37,72,32,67,8,22,59,109,30,66,5,19)(2,10,29,56,15,43,86,73,38,92,88,122,91,106,49,54,13,41,84,107,51,98,50,55,14,42,85,71,36,53,12,16)(4,18,58,108,27,64,87,93,34,69,99,126,89,120,80,117,28,65,112,128,82,119,79,116,26,63,111,127,78,115,25,62)(11,40,83,104,46,96,100,103,45,95,125,118,81,113,75,61,24,74,77,114,76,123,101,105,47,97,124,121,90,102,44,52);(1,4)(2,11)(3,18)(5,25)(6,34)(7,27)(8,26)(9,28)(10,40)(12,44)(13,24)(14,47)(15,46)(16,52)(17,58)(19,62)(20,69)(21,64)(22,63)(23,65)(29,83)(30,78)(31,87)(32,79)(33,80)(35,89)(36,90)(37,82)(38,45)(39,93)(41,74)(42,97)(43,96)(48,99)(49,75)(50,101)(51,76)(53,102)(54,61)(55,105)(56,104)(57,108)(59,111)(60,112)(66,115)(67,116)(68,117)(70,120)(71,121)(72,119)(73,103)(77,84)(81,91)(85,124)(86,100)(88,125)(92,95)(94,126)(98,123)(106,113)(107,114)(109,127)(110,128)(118,122)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320480, #72320861, #72320880, … |
| maximal normal subgroup of | #72320480, #72320861, #72320880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59491);