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 | |
6932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(3,8)(18,21)(22,24);(1,2)(3,4)(5,6)(7,8)(9,10,14,13)(11,12,16,15)(19,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24);(1,3,5,8)(2,7,6,4)(9,12,14,15)(10,16,13,11)(17,19)(18,22)(20,23)(21,24);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24);(1,4,5,7)(2,3,6,8)(9,13,14,10)(11,12,16,15)Polycyclic presentation:
2^9|b^2=i,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233544, #72233561, #72233608, … |
| maximal normal subgroup of | #72233544, #72233561, #72233608, … |
| maximal quotient of | #72235996, #72235998, #72297822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8632056);