Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150443 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C23 | C2×#99 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 56 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)(3,7)(4,6)(5,8)(9,10,13,18,15,20,12,27)(11,21,24,33,26,34,40,16)(14,19,28,35,31,37,30,17)(22,36,29,38,25,39,23,32);(2,6)(3,4)(7,8)(9,11,15,26)(10,16)(12,23,13,29)(14,22,31,25)(17,34)(18,36)(19,38)(20,33)(21,35)(24,28,40,30)(27,39)(32,37);(1,3,5,4)(2,7,8,6)(11,22)(14,31)(16,32)(17,35)(19,37)(21,36)(23,40)(24,29)(25,26)(28,30)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=ghi,d^2=gi,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfhi,d^b=dg,e^b=efg,f^b=fi,d^c=dhi,f^c=fi,e^d=ehi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57701);