Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1434 |
| prime radical | C24 | #117 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
502 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28,32,43)(20,33)(21,26)(22,36)(23,27)(24,30,39,31)(25,34)(29,38,45,48)(35,40)(37,41)(42,46,47,44);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24,27,39)(20,29)(21,30)(22,31)(23,32)(25,38,41,48)(26,28,36,43)(33,47,40,42)(34,44)(35,45)(37,46);(3,9)(6,12)(10,15)(13,16)(17,20,21,34,23,35,22,37)(18,25,26,40,27,41,36,33)(19,29,30,44,32,45,31,46)(24,38,28,42,39,48,43,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gh,c^b=ci,f^b=fh,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15627);