Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485649 |
| 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 | C22×#41 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
785 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 785 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)(19,23)(20,30)(21,33)(22,26)(24,32)(25,35)(27,42)(28,40)(29,38)(31,41)(34,39)(36,44)(37,46)(43,47)(45,48);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19,22,29)(18,23,26,38)(20,27,31,43)(21,28,35,46)(24,36,39,48)(25,40,33,37)(30,47,41,42)(32,44,34,45);(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,41)(26,39)(28,45)(29,43)(30,33)(32,35)(37,47)(38,48)(40,42)(44,46);(3,10)(6,15)(9,16)(11,13)(20,31)(24,39)(27,43)(30,41)(32,34)(36,48)(42,47)(44,45)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,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, 392926);