Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163726 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22)(25,26)(27,32)(28,37)(29,34)(30,39)(31,35)(33,38)(36,40);(1,3)(2,5)(4,8)(6,7)(25,27,31,28)(26,32,35,37)(29,38,30,36)(33,39,40,34);(3,8)(5,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,28,31,27)(26,33,35,40)(29,38,30,36)(32,39,37,34);(2,6)(5,7)(25,29,31,30)(26,34,35,39)(27,36,28,38)(32,40,37,33)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=i,g^2=hi,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, 120915);