Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,14)(12,19)(15,20)(17,23);(9,10,12,16,13,14,19,21)(11,15,17,22,18,20,23,24);(2,4)(5,6,8,7)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24);(5,7,8,6)(10,14)(12,19)(15,20)(17,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=g,h^2=i,b^a=be,c^a=cf,h^a=hi,d^b=de,e^b=ei,h^d=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263814);