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 | #18122 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 991 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,21)(12,24)(13,18)(14,19)(15,20)(16,23)(17,26)(22,35)(25,40)(27,33)(28,32)(29,34)(30,36)(31,38)(37,39);(1,2)(3,6)(4,8)(5,7)(9,11,15,13)(10,16,20,32)(12,22,26,27)(14,23,30,28)(17,31,24,39)(18,19,21,36)(25,37,29,38)(33,40,35,34);(1,3,5,4)(2,6,7,8)(9,12)(10,17,20,24)(11,22)(13,27)(14,29)(15,26)(16,31,32,39)(18,35,21,33)(19,34,36,40)(23,38)(25,30)(28,37);(9,13,15,11)(10,18,20,21)(12,22,26,27)(14,28,30,23)(16,19,32,36)(17,33,24,35)(25,37,29,38)(31,40,39,34)Polycyclic presentation:
2^9|b^2=i,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, 391979);