Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| 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 | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1115 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1115 conjugacy classes of subgroups — too many to draw. Only its 134 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:
(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,15,20,29)(12,16,24,32)(13,18,27,34)(19,37,38,28)(21,30,25,35)(22,31,39,40)(23,36,26,33);(2,5)(6,8)(9,11)(10,15)(12,22)(13,25)(14,20)(16,31)(17,29)(18,35)(19,23)(21,27)(24,39)(26,38)(28,33)(30,34)(32,40)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,32)(18,36)(20,38)(21,39)(22,25)(23,27)(29,37)(30,40)(31,35)(33,34);(2,5)(6,8)(10,17)(11,20)(16,32)(18,34)(19,38)(21,25)(22,39)(33,36)Polycyclic presentation:
2^9|a^2=i,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407385);