Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485337 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#120 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 48 points:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18,22,25)(19,27,29,41)(20,31,32,24)(21,34,37,26)(23,35,39,30)(28,43,45,42)(33,40,36,48)(38,44,47,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19,22,29)(18,23,25,39)(20,28,32,45)(21,35,37,30)(24,38,31,47)(26,41,34,27)(33,46,36,44)(40,43,48,42);(17,20,22,32)(18,24)(19,28,29,45)(21,36,37,33)(23,38)(25,31)(26,40)(27,42)(30,46,35,44)(34,48)(39,47)(41,43);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(18,25)(19,29)(20,32)(26,34)(30,35)(33,36)(38,47)(42,43)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392776);