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×#12186 |
| 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 | |
2041 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 2041 conjugacy classes of subgroups — too many to draw. Only its 138 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);(4,8)(5,6)(9,10)(11,12)(13,15)(14,17)(16,18);(3,4)(5,8)(6,10)(7,9)(12,15)(16,17);(3,5,7,6)(4,8,9,10)(11,13,14,18)(12,16)(15,17)56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339646, C2×#485608 |
| maximal normal subgroup of | #72339646, C2×#485608 |
| maximal quotient of | #72339753 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391958);