Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519063 |
| 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 | |
2105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,21)(13,19)(14,23)(15,18)(17,22)(20,24);(2,6)(7,8)(9,11,15,20)(10,16,18,24)(12,22,19,14)(13,23,21,17);(1,3,5,4)(2,7,6,8)(9,12,14,11,15,19,22,20)(10,16,17,21,18,24,23,13);(1,4,5,3)(2,8,6,7)(9,13,14,16,15,21,22,24)(10,11,17,19,18,20,23,12);(2,6)(7,8)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468971);