Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#12148 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1783 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1783 conjugacy classes of subgroups — too many to draw. Only its 146 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);(3,4)(5,7)(6,9)(8,10)(11,12)(13,15)(14,17)(16,18);(4,7)(5,10)(8,9)(12,15)(16,17);(11,13,14,18)(12,16)(15,17)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486255, C2×#485315, C2×#485933, … |
| maximal normal subgroup of | C2×#486255, C2×#485315, C2×#485933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391774);