Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485343 |
| 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 | #3496 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 820 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,26)(19,32)(20,31)(21,29)(23,37)(27,34)(28,35)(30,38)(33,36)(39,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,18,29)(12,19,23,33)(13,20,26,35)(16,27,30,39)(17,31,25,28)(22,36,37,32)(24,34,38,40);(9,12,14,23)(10,16)(11,19,21,33)(13,24,26,38)(15,27)(17,22)(18,30)(20,34,35,40)(25,37)(28,32)(29,39)(31,36);(1,3,5,4)(2,6,7,8)(12,23)(16,30)(19,33)(22,37)(24,38)(27,39)(32,36)(34,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392648);