Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| 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 | |
2207 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)(11,19)(12,22)(13,24)(14,18)(15,17)(16,21)(20,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,13)(14,21)(15,23)(17,22)(18,19)(20,24);(9,12,15,11,14,23,13,21)(10,16,24,20,18,19,17,22);(2,6)(5,7)(9,13,14,15)(10,17,18,24)(11,12,21,23)(16,22,19,20);(3,8)(5,7)(9,10)(11,20)(12,16)(13,17)(14,18)(15,24)(19,23)(21,22)Polycyclic presentation:
2^9|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,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589832);