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 | |
7185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,13,12)(11,15,16,14)(18,21)(22,24);(1,2)(3,4)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(19,23)(22,24);(2,6)(4,7)(10,12)(14,15)(17,18)(19,22)(20,21)(23,24);(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(1,4,5,7)(2,3,6,8)(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232278, #72232554, #72241230, … |
| maximal normal subgroup of | #72232278, #72232554, #72241230, … |
| maximal quotient of | #72232362, #72232628, #72232660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8137924);