Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7139 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C22 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #336 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(11,12,13,17,14,18,20,24,15,16,19,22,21,23,25,26);(3,4,5,7,6,8,9,10)(12,16)(13,19)(18,23)(20,25)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72210666 |
| maximal normal subgroup of | #72210666 |
| maximal quotient of | C2×#93250, C2×#92909, C2×#92910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33750);