Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133080 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1055 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,6,4,7)(5,8)(9,10)(11,19,22,31)(12,23,25,16)(13,17)(14,18)(15,21,29,26)(20,34)(24,38,40,30)(27,32)(28,35)(33,36)(37,39);(2,6)(3,4)(7,8)(9,11,14,22)(10,15,18,29)(12,20,25,36)(13,26,27,21)(16,28,23,37)(17,31,32,19)(24,39,40,35)(30,34,38,33);(1,3,5,4)(2,7,8,6)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,33)(21,35)(22,36)(26,39)(27,40)(29,37)(31,34)(32,38)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40407);