Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 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)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15);(1,3)(2,5)(4,8)(6,7)(17,18,23,24)(19,20,21,22);(9,11,13,12)(10,14,15,16)(17,19,22,18,23,21,20,24);(2,6)(5,7)(17,20,23,22)(18,21,24,19);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21,22,24,23,19,20,18)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6741659);