Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346727 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(14,20)(21,24);(1,2,5,7)(3,6,4,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(3,4)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(1,3,5,4)(2,6,7,8);(2,7)(6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253952);