Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150592 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11798 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| prime radical | #2018 | C22 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2923 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10)(3,13,17,27)(5,12,20,28)(6,11,19,29)(7,21,22,30)(8,18,24,16)(9,25,26,15)(14,23,31,32);(2,7)(3,14)(4,18)(5,16)(8,13)(9,12)(10,27)(11,23)(15,30)(19,20)(21,31)(22,28)(24,32)(26,29);(2,8,11,12)(3,15)(4,19)(7,13,23,9)(10,28,29,24)(14,30)(18,20)(22,26,32,27)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=dg,e^b=efg,g^b=gi,d^c=di,e^c=ei,e^d=eh,f^e=fi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57788);