Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2054 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)(3,6)(4,8)(5,7)(9,10)(11,16)(12,22)(13,20)(14,18)(15,19)(17,23)(21,24);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,23)(13,18)(14,20)(15,21)(17,22)(19,24);(9,12,14,10,15,18,22,19)(11,16,17,23,21,24,20,13);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,7,3,8)(9,13,14,16,15,23,22,24)(10,17,18,21,19,20,12,11)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718700);