Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#11796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
3796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,17,18,15)(14,16);(4,7)(5,10)(8,9)(12,15)(16,17);(4,8)(7,9)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238070, C2×#482360, #72240372, … |
| maximal normal subgroup of | #72238070, C2×#482360, #72240372, … |
| maximal quotient of | C2×#129729, #72261843, C22×#11796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389007);