Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #331 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7102 |
| prime radical | #532 | C8 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,4,10,6,13,22,37,5,9,21,32,24,33,50,58,7,12,23,36,27,40,52,61,25,11,38,35,53,39,63,60)(3,14,15,42,17,43,45,62,18,8,20,28,48,31,49,57,19,29,46,34,26,54,55,59,16,30,44,56,47,41,51,64);(1,3)(2,8)(4,20)(5,18)(6,26)(7,19)(9,29)(10,34)(11,14)(12,30)(13,41)(15,38)(16,25)(17,27)(21,46)(22,51)(23,44)(24,47)(28,35)(31,40)(32,56)(33,43)(36,42)(37,62)(39,54)(45,50)(48,53)(49,52)(55,63)(57,58)(59,61)(60,64)Polycyclic presentation:
2^9|a^2=d,c^2=f,d^2=g,e^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=defi,e^b=ei,g^b=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1085);