Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#13088 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4)(5,6)(7,8)(9,13,14,10)(11,12);(7,9,13,12)(8,11,14,10);(4,6)74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232683, #72232728, C2×#499787, … |
| maximal normal subgroup of | #72232683, #72232728, C2×#499787, … |
| maximal quotient of | #72232651, #72232719, #72232743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401904);