Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13058 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2010 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 2010 conjugacy classes of subgroups — too many to draw. Only its 258 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);(4,6)(7,8,9,13,12,17,10,15)(11,16,19,14,20,22,18,21);(3,4,5,6)(8,13)(9,10)(11,20)(14,22)(15,17)(16,21);(7,9,12,10)(8,14,17,21)(11,19,20,18)(13,16,15,22)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266753, #72266755, #72266853, … |
| maximal normal subgroup of | #72266753, #72266755, #72266853, … |
| maximal quotient of | C2×#148173, C2×#147778, C2×#147496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401811);