Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,15)(13,17)(14,18)(19,21)(20,23)(22,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,19)(12,17)(13,15)(14,18)(16,21)(20,24)(22,23);(10,15)(12,13)(16,19)(17,18)(21,22)(23,24);(1,3,5,4)(2,6,7,8)(11,20)(16,23)(19,24)(21,22);(1,4,5,3)(2,6,7,8)(9,11,14,20)(10,16,18,23)(12,21,13,22)(15,19,17,24)Polycyclic presentation:
2^9|a^2=f,d^2=f,e^2=fi,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,g^c=gi,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, 6583713);