Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133904 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2316 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
699 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 699 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2,3,6,4,5,7,8)(9,10)(11,19)(12,23,24,34)(13,26,29,17)(14,18)(15,21)(16,25)(20,36,38,40)(22,31,27,33)(28,32)(30,39)(35,37);(2,5)(3,7)(9,11)(10,15,18,19)(12,20)(13,27)(14,21)(16,30,32,35)(17,33,26,31)(22,29)(23,36,34,40)(24,38)(25,39)(28,37);(9,12)(10,16)(11,20)(13,28)(14,24)(15,30)(17,34)(18,32)(19,35)(21,38)(22,39)(23,26)(25,29)(27,37)(31,36)(33,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fhi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41371);