Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1408 |
| commutator | #376 | C2×C4 |
| Frattini, cosocle | #1963 | C22 |
| socle | C2 | #7434 |
| 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:
(1,2,5,11)(3,14,16,46)(4,19,21,42)(6,13,27,45)(7,12,26,43)(8,28,30,44)(9,31,32,20)(10,36,38,22)(15,33,49,41)(17,37,52,59)(18,48,51,62)(23,40,25,57)(24,39,56,63)(29,47,53,64)(34,58,60,50)(35,55,61,54);(1,3,7,17,6,18,8,29)(2,9,13,35)(4,20,23,15,24,54,22,50)(5,16,26,52,27,51,30,53)(10,37)(11,41)(12,34,28,33)(14,19)(21,55,25,58,56,31,36,49)(32,43)(38,62,57,46)(39,48)(40,47)(42,59,63,64)(44,61)(45,60)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=fghi,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=defgi,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, 1916);