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 | |
6894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(17,18)(19,23)(20,24)(21,22);(3,8)(6,7)(9,10,13,12)(11,15,16,14)(18,22)(19,24);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,24)(22,23);(3,8)(6,7)(11,16)(14,15)(17,20)(18,19)(21,23)(22,24);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,22)(23,24);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,14,16,15)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72233860, #72262009, #72262018, … |
| maximal normal subgroup of | #72233860, #72262009, #72262018, … |
| maximal quotient of | #72233996, #72234006, #72263367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935052);