Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60591 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,19)(13,16,20,24)(14,17,22,18)(21,23,26,25);(4,7)(8,10)(11,13)(12,16)(14,21)(15,20)(17,23)(18,25)(19,24)(22,26);(12,17)(14,22)(16,23)(18,19)(21,26)(24,25);(3,5,6,9)(4,8,7,10)(11,13)(12,16)(14,21)(15,20)(17,23)(18,25)(19,24)(22,26);(13,20)(16,24)(21,26)(23,25)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482322);