Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | #2150 | C22 |
| socle | C2 | #7056 |
| prime radical | #532 | C8 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,10,6,12,23,36,5,9,24,33,25,34,52,58,7,13,21,37,27,40,49,60,26,11,22,38,50,39,51,61)(3,14,15,35,17,41,43,59,18,8,28,29,45,31,53,54,19,30,42,55,46,56,62,64,16,32,20,47,44,57,48,63);(1,3)(2,8)(4,20)(5,18)(6,17)(7,19)(9,30)(10,35)(11,14)(12,31)(13,32)(15,24)(16,26)(21,28)(22,42)(23,48)(25,45)(27,46)(29,33)(34,56)(36,59)(37,55)(38,47)(39,41)(40,57)(43,52)(44,50)(49,53)(51,62)(54,58)(60,64)(61,63)Polycyclic presentation:
2^9|a^2=d,c^2=g,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cg,d^b=degi,e^b=ei,d^c=dgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 889);