Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
637 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 637 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,9,14,11,17,30,16)(7,18,20,47,23,48,22,54)(8,24,27,55,29,57,62,56)(10,31,26,15,33,43,61,45)(12,34,35,50,37,49,46,19)(13,38,40,63,42,65,41,28)(21,52,51,66,53,67,32,39)(25,44,58,64,60,68,59,36);(5,7)(6,12)(8,25)(9,20)(10,32)(11,23)(13,39)(14,35)(15,44)(16,46)(17,37)(18,24)(19,28)(21,26)(22,30)(27,58)(29,60)(31,36)(33,51)(34,38)(40,52)(41,67)(42,66)(43,64)(45,68)(47,55)(48,57)(49,65)(50,63)(53,61)(54,56)(59,62);(2,4)Polycyclic presentation:
2^9|a^2=f,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59356);