Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163726 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24)(27,33,34,35)(30,37,38,40);(1,3,4,8)(2,5,6,7)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,38)(35,40)(36,39);(3,8)(5,7)(10,15)(12,19)(14,20)(17,23)(25,27)(26,30)(28,35)(29,34)(31,38)(32,40)(33,36)(37,39);(2,6)(5,7)(26,31)(30,38)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=eh,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122393);