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 | |
6832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(4,8)(6,7)(10,14)(11,16)(18,21)(19,20);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18,22,21)(19,24,20,23);(2,3)(6,7)(11,16)(12,15)(17,19,22,20)(18,23,21,24);(1,2,5,3)(4,6,8,7)(9,11,13,16)(10,15,14,12)(17,20,22,19)(18,24,21,23);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,16,14,11)(17,21,22,18)(19,24,20,23);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=ghi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232577, #72297212 |
| maximal normal subgroup of | #72232577, #72297212 |
| maximal quotient of | #72232605, #72297229, #72297646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038700);