Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574063 |
| 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 | |
1825 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)(4,8)(5,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,22)(23,24);(2,6)(7,8)(9,10)(11,19)(12,16)(13,18)(14,17)(15,21)(20,23)(22,24);(1,3,5,4)(2,7,6,8)(9,10,14,17,13,18,12,16)(11,15,22,19,23,24,21,20);(1,4,5,3)(2,8,6,7)(9,10)(11,20)(12,16)(13,18)(14,17)(15,22)(19,23)(21,24);(9,11)(10,15)(12,21)(13,23)(14,22)(16,20)(17,19)(18,24)Polycyclic presentation:
2^9|c^2=fgi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492018);