Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485065 |
| 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 | #3496 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
840 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 840 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,26)(14,17)(15,21)(18,29)(20,36)(22,35)(23,39)(24,34)(25,40)(27,31)(28,33)(30,37)(32,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,17,31)(12,23,24,20)(13,27,29,21)(16,32,34,30)(18,35,26,19)(25,37,28,38)(33,36,40,39);(9,12,14,24)(10,16,17,34)(11,20,22,23)(13,28,29,25)(15,30,31,32)(18,33,26,40)(19,36,35,39)(21,38,27,37);(1,3,5,4)(2,6,7,8)(10,17)(12,24)(15,31)(18,26)(19,35)(20,23)(25,28)(37,38)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,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, 392398);