Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1889 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 535 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,6,7,5,3,4,8)(9,10)(11,20,25,36)(12,17)(13,18,29,34)(14,19)(15,23,33,22)(16,26,35,28)(21,30)(24,31)(27,40)(32,38)(37,39);(1,3,5,2)(4,8,6,7)(9,11,21,23,14,25,12,22)(10,15,30,20,19,33,17,36)(13,27,28,38,29,24,26,39)(16,31,34,37,35,40,18,32);(1,4,5,6)(2,7,3,8)(9,12,14,21)(10,16,19,35)(11,22,25,23)(13,28,29,26)(15,31,33,40)(17,18,30,34)(20,37,36,32)(24,39,27,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,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, 44366);