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 | |
6198 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)(17,18,21,20)(19,22,24,23);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,16,12)(17,18)(19,22)(20,21)(23,24);(2,5)(6,7)(10,14)(12,15)(17,19,21,24)(18,22,20,23);(3,8)(6,7)(10,14)(11,16)(18,20)(22,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20,21,18)(19,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232519, #72281380, #72326382 |
| maximal normal subgroup of | #72232519, #72281380, #72326382 |
| maximal quotient of | #72232598, #72282308, #72282380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8294778);