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 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
494 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 494 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) = 128 |
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)(19,29,32,39)(20,34)(21,26)(22,27)(23,28)(24,31,40,33)(25,36)(30,44,47,38)(35,41)(37,42)(43,45,48,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,33,23,32,21,31)(18,24,26,29,28,40,27,39)(20,30,36,46,37,47,35,45)(25,38,34,43,41,44,42,48);(3,9)(6,12)(10,15)(13,16)(17,20,21,35,23,37,22,36)(18,25,26,34,28,41,27,42)(19,30,31,45,32,47,33,46)(24,38,29,43,40,44,39,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=i,g^2=h,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, 15629);