Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136871 |
| 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)(3,9,18,32,16,33,19,34)(4,20)(5,12,27,40,28,37,29,41)(10,26)(15,43)(17,31,49,59,47,60,25,38)(21,53)(22,42)(23,39)(24,52)(30,48)(35,55)(36,54)(44,51)(45,56)(46,62)(50,61)(57,63)(58,64);(1,3,8,18,7,16,6,19)(2,9,13,34,11,33,14,32)(4,15,21,44,24,46,23,50)(5,25,29,17,28,49,27,47)(10,30,35,56,36,58,42,57)(12,38,40,60,37,59,41,31)(20,51,39,43,52,61,53,62)(22,48,54,63,55,64,26,45);(1,4)(2,10)(3,15)(5,26)(6,23)(7,24)(8,21)(9,30)(11,36)(12,39)(13,35)(14,42)(16,46)(17,48)(18,44)(19,50)(20,41)(22,29)(25,45)(27,55)(28,54)(31,51)(32,57)(33,58)(34,56)(37,53)(38,43)(40,52)(47,64)(49,63)(59,62)(60,61)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=eh,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, 44067);