Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,22)(21,24);(1,2)(3,6)(4,7)(5,8)(17,19)(18,21)(20,24)(22,23);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(18,22)(21,23);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,5)(4,7)(6,8)(9,11,12,16)(10,14,13,15)(17,19)(18,21)(20,24)(22,23);(1,3)(2,6)(4,8)(5,7)Polycyclic presentation:
2^9|a^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72232195, #72232384, #72232577, … |
| maximal normal subgroup of | #72232195, #72232384, #72232577, … |
| maximal quotient of | #72232262, #72232413, #72232566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7822710);