Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1799 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 86 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)(19,27)(20,31)(21,34,35,24)(22,26)(23,29)(25,33)(28,42)(30,40,44,41)(32,39,47,46)(36,48,37,45)(38,43);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,29,22,28,33,43)(18,23,25,27,26,38,31,42)(21,30,32,45,35,44,47,48)(24,37,39,41,34,36,46,40);(3,9)(6,12)(10,15)(13,16)(18,24,26,34)(19,28)(21,35)(23,36,38,37)(25,39,31,46)(27,40,42,41)(29,43)(32,47)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=chi,e^c=ei,f^c=fg,f^e=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41545);