Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#12852 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17,18,15)(14,16);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,15)(16,17)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#496228, #72234790, #72234793, … |
| maximal normal subgroup of | C2×#496228, #72234790, #72234793, … |
| maximal quotient of | #72236988, C2×#152324, C2×#152322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401048);