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 | |
6443 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)(3,7)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,22,20)(19,24,21,23);(1,2,5,4)(3,7,8,6)(17,19,22,21)(18,23,20,24);(1,3)(2,6)(4,7)(5,8)(10,11)(14,15)(18,20)(23,24);(1,2,5,4)(3,7,8,6)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,24,21,23);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,21,22,19)(18,24,20,23);(1,3)(2,7)(4,6)(5,8)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=h,e^2=g,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 | #72232384, #72234852, #72234858, … |
| maximal normal subgroup of | #72232384, #72234852, #72234858, … |
| maximal quotient of | #72232439, #72326557, #72326593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7806434);