Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136422 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8894 |
| 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 | |
318 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 318 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)(3,8,15,32)(4,18)(5,23)(6,10)(7,28)(9,20)(11,37)(12,13)(14,43,46,64)(16,48,47,38)(17,41,29,33)(19,52)(21,39)(22,53)(24,57,59,31)(25,34)(26,40)(27,36)(30,58,61,45)(35,56)(42,55)(44,63,50,51)(49,62,54,60);(1,3,13,17,6,15,7,29)(2,8,28,33,10,32,12,41)(4,14,21,50,22,46,19,44)(5,24,37,47,26,59,27,16)(9,30,34,60,35,61,42,62)(11,38,23,57,36,48,40,31)(18,51,52,43,53,63,39,64)(20,54,55,45,56,49,25,58);(1,4)(2,9)(3,14)(5,25)(6,22)(7,19)(8,30)(10,35)(11,39)(12,42)(13,21)(15,46)(16,49)(17,50)(18,23)(20,37)(24,58)(26,55)(27,56)(28,34)(29,44)(31,63)(32,61)(33,60)(36,52)(38,64)(40,53)(41,62)(43,48)(45,59)(47,54)(51,57)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,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, 43618);