Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12181 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
949 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 949 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) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7)(5,9)(8,10)(11,12)(13,16)(14,17)(15,19)(18,23)(20,24)(21,26)(22,25);(3,4,6,8)(5,10,9,7)(11,13,15,21)(12,16,19,26)(14,22,23,20)(17,24,18,25);(12,17,19,18)(16,24,26,25)62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#486416, C2×#485793, C2×#485442 |
| maximal normal subgroup of | C2×#486416, C2×#485793, C2×#485442 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391906);