Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C8 · Schur multiplier (H2) = C212 |
| direct product | C23×#336 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C4 | C22 |
| commutator | C2 | C23×C4×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle, prime radical | C26 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
12018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4);(5,6);(7,8)(9,10)(15,16,17,19,18,20,21,22);(8,10)(11,12,13,14)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C23×#1879, C23×#1864, C23×#1749, … |
| maximal normal subgroup of | C23×#1879, C23×#1864, C23×#1749, … |
| maximal quotient of | C23×#1375 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276927);