Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152136 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12707 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #536 | C23 |
| prime radical | #61645 | C2 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3486 ◁ G | |
| LECS | G ▷ #536 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 643 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 64 points:
(1,2,5,10,7,12,25,39)(3,13,16,44,14,43,46,61)(4,18,21,52,20,51,55,64)(6,11,27,41,28,42,24,38)(8,29,31,48,30,57,60,54)(9,33,35,22,34,58,37,56)(15,32,47,36,19,40,53,62)(17,45,49,63,26,50,23,59);(1,3)(2,8)(4,19)(5,23)(6,26)(7,14)(9,13)(10,36)(11,40)(12,30)(15,20)(16,24)(17,28)(18,50)(21,57)(22,48)(25,49)(27,46)(29,55)(31,41)(32,42)(33,53)(34,43)(35,63)(37,59)(38,60)(39,62)(44,52)(45,51)(47,58)(54,56)(61,64);(3,14)(4,20)(8,30)(13,43)(16,46)(17,26)(21,55)(22,56)(23,49)(31,60)(32,40)(33,58)(36,62)(44,61)(45,50)(59,63)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fhi,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59338);