Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#10935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C27 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
10942 conjugacy classes of subgroups; only the 778 normal subgroups are shown.
This group has 10942 conjugacy classes of subgroups — too many to draw. Only its 778 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,6)(11,12)(13,17,18,15)(14,16);(7,8,9,10)(11,13)(12,15)(14,18)(16,17);(3,4,5,6)(11,12)(13,15)(14,16)(17,18)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#364410, C2×#417744, #72221174, … |
| maximal normal subgroup of | C2×#364410, C2×#417744, #72221174, … |
| maximal quotient of | #72266011, C2×#147497, C2×#110401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261224);