Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13359 |
| 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 | |
2154 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 2154 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,12,13)(9,16,19,22)(10,20,15,11)(14,17,21,18);(7,9)(8,14)(10,16)(11,17)(12,19)(13,21)(15,22)(18,20);(4,6)(7,10,12,15)(8,11,13,20)(9,17,19,18)(14,22,21,16)62 conjugacy classes — view the full character table.
| maximal subgroup of | #72266762, #72266765, #72266852, … |
| maximal normal subgroup of | #72266762, #72266765, #72266852, … |
| maximal quotient of | C2×#148173, C2×#147778, #72267374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406098);