Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7213 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12)(13,17,18,15)(14,16);(4,7,9,8)(12,15)(16,17)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#135486, C2×#11710955, C2×#139504, … |
| maximal normal subgroup of | C2×#135486, C2×#139504, C2×#127199, … |
| maximal quotient of | #72261719, C22×#7213, #72291969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34379);