Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141707 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1961 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19661 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
412 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10)(11,22,27,37)(12,29,31,36)(13,32)(14,19)(15,21)(16,25,35,17)(18,23)(20,34)(24,33)(26,39,28,30)(38,40);(9,11,23,36,34,26,13,17,15,27,40,29,14,28,33,25)(10,16,32,30,19,12,18,37,21,35,24,39,20,31,38,22);(1,2,6,8,4,7,3,5)(9,12)(10,17,21,25)(11,24,27,32)(13,30)(14,16)(15,31)(18,26,38,28)(19,29,20,36)(22,23)(33,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eghi,d^2=g,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224887 |
| maximal normal subgroup of | #72224887 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48932);