Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485643 |
| 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×C4⋊C4 | C24 |
| prime radical | #537 | C23 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
709 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 709 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) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18)(19,23)(20,25)(21,24)(22,27)(26,35)(28,37)(29,36)(30,39)(31,41)(32,40)(33,43)(34,38)(42,46)(44,47)(45,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(17,19,22,30)(18,23,27,39)(20,31,33,28)(21,34,35,29)(24,36,26,38)(25,41,43,37)(32,44,46,45)(40,48,42,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(18,24)(21,35)(23,36)(25,40)(26,27)(29,34)(32,46)(37,47)(38,39)(41,48)(42,43)(44,45);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(17,20,22,33)(18,25,27,43)(19,28,30,31)(21,32,35,46)(23,37,39,41)(24,40,26,42)(29,44,34,45)(36,47,38,48)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=dgi,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393232);