Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485611 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #2815 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
810 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18,22,27)(19,24,28,40)(20,25,33,43)(21,23,34,26)(29,39,31,41)(30,36,35,38)(32,37,46,42)(44,47,45,48);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(18,23)(19,28)(21,34)(24,38)(25,37)(26,27)(29,31)(32,46)(36,40)(39,48)(41,47)(42,43);(17,19,22,28)(18,24,27,40)(20,31,33,29)(21,35,34,30)(23,36,26,38)(25,41,43,39)(32,44,46,45)(37,48,42,47);(3,10)(6,15)(9,16)(11,13)(17,20,22,33)(18,25,27,43)(19,29,28,31)(21,32,34,46)(23,37,26,42)(24,39,40,41)(30,44,35,45)(36,47,38,48)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392911);