Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#12123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1655 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1655 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) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26);(3,4,6,8)(5,10,9,7)(11,13)(12,16)(14,21)(15,20)(17,19)(18,23)(22,26)(24,25);(13,20)(14,22)(16,23)(19,25)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#485605, #72284405, #72284431, … |
| maximal normal subgroup of | C2×#485605, #72284405, #72284431, … |
| maximal quotient of | #72285537, C2×#151196, #72292045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391473);