Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C8×#453 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×#453 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,5,4,6,7,8);(10,13)(11,15)(14,16);(9,10)(11,16)(12,14)(13,15);(11,15)(13,16)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216435, C16×#453, #453×#3633, … |
| maximal normal subgroup of | #72216435, C16×#453 |
| maximal quotient of | #72216491, C16×#453, #453×#3633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 115936);