Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #418 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,5,7,4,3,8,6)(9,10,13,19,16,21,15,33)(11,22,25,17,12,26,23,35)(14,29,28,34,32,40,31,20)(18,30,37,24,38,39,36,27);(2,3)(6,7)(9,11)(10,17,21,35)(12,16)(13,23)(14,27)(15,25)(18,20,38,34)(19,22,33,26)(24,32)(28,39)(29,37,40,36)(30,31);(1,3)(2,4)(5,6)(7,8)(9,12,16,11)(10,18,21,38)(13,23,15,25)(14,30,32,39)(17,34,35,20)(19,37,33,36)(22,29,26,40)(24,31,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=dg,c^2=i,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41612);