Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C2×#1598 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1083 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1083 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) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,8)(5,6)(9,10)(11,12)(13,19)(14,22)(15,18)(16,21)(17,24)(20,26)(23,25);(3,4)(5,8)(6,10)(7,9)(17,22)(19,26);(3,5,7,6)(4,8,9,10)(11,13,15,20)(12,16,18,25)(14,23,24,21)(17,26,22,19)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#485793, C2×#485206, C2×#486533 |
| maximal normal subgroup of | C2×#485793, C2×#485206, C2×#486533 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391575);