Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29751 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1176 conjugacy classes of subgroups.
| 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)(11,20,24,23)(12,25)(16,21,30,33)(17,31)(22,36,37,34)(26,27)(29,39,40,35)(32,38);(2,5)(6,8)(9,10)(11,21)(12,17)(13,18)(14,28)(15,19)(16,23)(20,30)(22,35)(24,33)(25,31)(26,32)(27,38)(29,36)(34,40)(37,39);(3,7)(6,8)(9,11)(10,16)(12,22)(13,23)(14,20)(15,24)(17,29)(18,33)(19,30)(21,28)(25,37)(26,36)(27,34)(31,40)(32,39)(35,38);(1,3,4,7)(2,6,5,8)(9,12,13,26,15,25,14,27)(10,17,18,32,19,31,28,38)(11,22,23,36,24,37,20,34)(16,29,33,39,30,40,21,35)Polycyclic presentation:
2^9|a^2=h,d^2=efh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289070);