Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,23)(13,18)(14,19)(15,20)(17,22)(21,24);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,13)(14,24)(15,22)(17,20)(18,23)(19,21);(9,12,14,11,15,24,13,22)(10,17,18,21,20,16,19,23);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,12,22,24)(16,23,17,21);(3,8)(5,7)(9,10)(11,21)(12,17)(13,19)(14,18)(15,20)(16,24)(22,23)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590456);