Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1889 |
| prime radical | #18122 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
613 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 613 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,3,8,7)(9,10,14,18)(11,19,21,38)(12,24,20,17)(13,26)(15,30,33,22)(16,29)(23,36)(25,35)(27,37)(28,34)(31,39)(32,40);(1,3)(2,5)(4,6)(7,8)(9,11,20,30,14,21,12,22)(10,15,24,38,18,33,17,19)(13,27,28,39,29,23,25,40)(16,32,34,36,26,31,35,37);(1,2)(3,5)(4,6)(7,8)(10,16)(11,21)(15,31)(17,35)(18,26)(19,36)(22,30)(23,27)(24,34)(32,33)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=di,e^a=egh,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44558);