Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163726 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 | #35292 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1059 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)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23)(25,26)(27,36)(28,35)(29,34)(30,33)(31,38)(32,37)(39,40);(1,3)(2,5)(4,8)(6,7)(25,27,29,38)(26,31,34,36)(28,39,30,37)(32,33,40,35);(3,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,32)(28,37)(29,38)(30,39)(31,35)(33,36)(34,40);(2,6)(5,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28,29,30)(26,33,34,35)(27,37,38,39)(31,32,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,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, 120901);