Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| 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 | |
138 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 138 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:
(1,2,4,10,6,12,22,36,5,11,24,35,25,38,51,59,7,13,23,37,27,39,50,60,26,9,21,32,48,33,49,57)(3,14,15,34,17,40,42,58,19,31,41,54,46,56,61,64,18,29,43,52,45,55,62,63,16,8,20,28,44,30,47,53);(1,3)(2,8)(4,20)(5,19)(6,17)(7,18)(9,29)(10,34)(11,14)(12,30)(13,31)(15,24)(16,26)(21,43)(22,47)(23,41)(25,46)(27,45)(28,32)(33,55)(35,54)(36,58)(37,52)(38,40)(39,56)(42,51)(44,48)(49,62)(50,61)(53,57)(59,64)(60,63)Polycyclic presentation:
2^9|a^2=d,c^2=gi,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=cgi,d^b=deg,e^b=ei,d^c=dgi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 890);