Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C22×Q8×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
12322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2);(3,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12);(14,16)(17,18,19,20);(13,14,15,16)200 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#59641, C22×#36114, C22×#59314, … |
| maximal normal subgroup of | C22×#59641, C22×#36114, C22×#59314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532538);