Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94646 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1409 |
| commutator | #376 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| socle | C2 | #7435 |
| prime radical | #532 | D4 |
| LCS | G ▷ #376 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 131 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)(3,14,16,45)(4,20,22,54)(6,11,28,42)(7,13,27,43)(8,31,32,21)(9,26,37,24)(12,29,44,30)(15,33,47,40)(17,46,25,58)(18,36,51,63)(19,52,53,60)(23,38,57,41)(34,59,61,49)(35,50,62,48)(39,56,64,55);(1,3,6,18,7,17,30,19)(2,8,13,35)(4,21,55,15,23,48,24,49)(5,25,28,53,27,16,29,51)(9,36)(10,40)(11,34,12,33)(14,20)(22,31,56,47,57,50,26,59)(32,42)(37,58,64,45)(38,46)(39,52)(41,60,54,63)(43,61)(44,62)Polycyclic presentation:
2^9|a^2=d,b^2=egh,c^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=defg,e^b=ei,g^b=gi,d^c=dghi,e^c=ei,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1922);