Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94523 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7338 |
| commutator | C22×Q8 | C2×C8 |
| Frattini, cosocle | #1782 | C22 |
| prime radical | #3487 | C4 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1782 ◁ G | |
| LECS | G ▷ #1782 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
280 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 64 points:
(1,2,5,10)(3,13,15,46,17,35,51,62)(4,18,20,53,22,44,55,63)(6,26,24,40)(7,12,25,41)(8,30,31,49)(9,34,36,28,37,58,39,21)(11,42,45,29)(14,43,48,38)(16,47,23,56,27,59,57,64)(19,32,50,61)(33,54,60,52);(1,3)(2,8)(4,19)(5,23)(6,27)(7,17)(9,35)(10,38)(11,43)(12,33)(13,37)(14,22)(15,24)(16,29)(18,47)(20,54)(21,49)(25,57)(26,32)(28,52)(30,55)(31,40)(34,50)(36,56)(39,64)(41,61)(42,51)(44,59)(45,60)(46,53)(48,58)(62,63)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1719);