Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C8 | C82 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7137 |
| prime radical | C23 | #347 |
| LCS | G ▷ C8 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,6,13,26,40,7,12,27,42,23,37,54,63,8,14,25,41,29,44,57,62,21,35,53,39,4,10,22,36)(3,15,17,46,18,34,52,61,19,43,24,55,16,45,49,59,20,47,51,64,28,9,30,31,48,33,56,60,50,32,58,38);(1,3)(2,9,12,32,14,34,35,45)(4,16)(5,24,25,56)(6,28)(7,19)(8,20)(10,15,13,43,37,47,44,33)(11,38,39,31,41,59,42,61)(17,27,51,53)(18,29)(21,48)(22,30,54,52)(23,50)(26,58,57,49)(36,55,62,46,63,60,40,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,e^a=ehi,g^a=gi,d^b=dh,f^b=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72210687, #72217187, #72217188, … |
| maximal normal subgroup of | #72210687, #72217187, #72217188, … |
| maximal quotient of | #72210682, #72227252, #72234264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1191);