Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7015 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C24 | #95 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,13,15,14,16,17,18);(4,7,9,8)(19,20,21,23,22,24,25,26)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#96892, C2×C8×#334, C2×#96686, … |
| maximal normal subgroup of | C2×#96892, C2×C8×#334, C2×#96686, … |
| maximal quotient of | C2×#93270, C2×#93269, C2×#93266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2074);