Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136874 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8977 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,6,13,7,11,8,14)(3,9,18,33,16,34,19,35)(4,20)(5,25,28,42,29,12,30,41)(10,36)(15,44)(17,49,51,39,48,32,26,58)(21,40)(22,43)(23,54)(24,55)(27,38)(31,56)(37,57)(45,53)(46,60)(47,64)(50,62)(52,63)(59,61);(1,3,8,18,7,16,6,19)(2,9,13,35,11,34,14,33)(4,15,21,45,24,47,23,52)(5,26,30,17,29,51,28,48)(10,31,37,60,38,62,43,61)(12,39,41,49,25,58,42,32)(20,53,54,44,55,63,40,64)(22,56,27,59,57,50,36,46);(1,4)(2,10)(3,15)(5,27)(6,23)(7,24)(8,21)(9,31)(11,38)(12,40)(13,37)(14,43)(16,47)(17,50)(18,45)(19,52)(20,41)(22,28)(25,54)(26,59)(29,36)(30,57)(32,63)(33,61)(34,62)(35,60)(39,64)(42,55)(44,58)(46,51)(48,56)(49,53)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44070);