Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1590 conjugacy classes of subgroups — too many to draw. Only its 186 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);(3,4,6,8)(5,9,10,7)(12,15)(13,17)(16,22)(19,21)(20,23)(25,26);(3,5,6,10)(4,7,8,9)(11,12,14,15)(13,19,20,26)(16,24,22,18)(17,25,23,21);(11,13)(12,17)(14,20)(15,23)(16,25)(18,26)(19,24)(21,22)62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210355, #72210373, #72212048, … |
| maximal normal subgroup of | #72210355, #72210373, #72212048, … |
| maximal quotient of | C2×#148173, C2×#148172, #72267536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406184);