Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C2×#135 | Q8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(17,18)(19,32)(20,25)(21,41)(22,30)(23,29)(24,31)(26,34)(27,38)(28,40)(33,44)(35,43)(36,42)(37,47)(39,46)(45,48);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(18,25)(20,38)(21,40)(22,23)(26,42)(27,29)(28,31)(30,41)(32,44)(33,34)(35,36)(39,48)(43,46)(45,47);(17,19,24,37)(18,26,31,46)(20,39,40,34)(21,33,38,48)(22,36,23,35)(25,42,28,43)(27,47,41,32)(29,45,30,44)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=gi,e^2=hi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43665);