Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484587 |
| 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 | #392 | C23 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
800 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18)(19,27)(20,25)(21,23)(22,26)(24,30)(28,40)(29,42)(31,39)(32,37)(33,41)(34,36)(35,38)(43,47)(44,46)(45,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,23)(24,36)(25,37)(26,38)(27,42)(39,46)(40,47)(41,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19)(18,24,26,27)(20,31)(21,34)(22,30)(23,36,38,42)(25,40,41,39)(28,33)(29,35)(32,43)(37,47,48,46)(44,45);(17,20,22,33)(18,25,26,41)(19,28,30,31)(21,32,35,45)(23,37,38,48)(24,39,27,40)(29,43,34,44)(36,46,42,47)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392034);