Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133835 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 718 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,27,23,28,19,24)(20,31,29,46,32,45,38,48)(21,34,30,44,37,26,35,40)(25,41,43,36,42,47,39,33);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,23,22)(18,24,28,27)(20,29,32,38)(21,35,37,30)(25,39,42,43)(26,44,34,40)(31,48,45,46)(33,41,36,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,36)(22,38)(23,32)(24,39)(26,45)(27,43)(28,42)(30,41)(31,34)(33,37)(35,47)(40,48)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,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, 41135);