Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11583 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,13,17,14,18,19,24,15,16,20,22,21,23,25,26);(4,7)(5,10)(8,9);(12,16)(17,22)(18,23)(24,26)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#140565, C2×#140595, C2×#481116, … |
| maximal normal subgroup of | C2×#140565, C2×#140595, C2×#481116, … |
| maximal quotient of | #72320926, #72321624, C2×#122474 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388237);