Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622979 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,26,46)(12,17,27,38)(13,30,32,58)(14,33,31,19)(16,34,37,24)(18,41,42,22)(23,43,49,65)(25,53,52,45)(28,56,54,71)(29,57,55,40)(35,59,61,50)(36,47,44,60)(39,64,63,48)(51,70,69,67)(62,68,66,72);(2,5)(6,8);(9,11,22,47,14,25,32,34,15,26,41,60,31,52,13,24)(10,16,30,45,19,36,42,46,20,37,58,53,33,44,18,21)(12,23,48,68,29,51,54,59,27,49,64,72,55,69,28,50)(17,35,56,67,40,62,63,65,38,61,71,70,57,66,39,43);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,23)(13,28)(14,29)(15,27)(16,35)(18,39)(19,40)(20,38)(21,43)(22,48)(24,50)(25,51)(26,49)(30,56)(31,55)(32,54)(33,57)(34,59)(36,62)(37,61)(41,64)(42,63)(44,66)(45,67)(46,65)(47,68)(52,69)(53,70)(58,71)(60,72);(12,27)(17,38)(23,49)(28,54)(29,55)(35,61)(39,63)(40,57)(43,65)(48,64)(50,59)(51,69)(56,71)(62,66)(67,70)(68,72)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530458);