Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,30,35)(27,36,39,48)(28,41,44,51)(29,33,37,34)(31,45,47,40)(32,49,52,43)(38,50,42,46)(53,56,55,54);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(25,27,37,45,30,39,29,40)(26,31,33,48,35,47,34,36)(28,42,49,54,44,38,43,56)(32,50,51,53,52,46,41,55);(11,18)(14,17)(19,24)(21,23)(25,28,29,43,30,44,37,49)(26,32,34,41,35,52,33,51)(27,38,40,54,39,42,45,56)(31,46,36,53,47,50,48,55);(1,4,5,3)(2,6,7,8)(12,20)(15,22)(19,24)(21,23)(27,39)(31,47)(36,48)(38,42)(40,45)(46,50)(53,55)(54,56);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,30,37)(26,33,35,34)(27,40,39,45)(28,43,44,49)(31,48,47,36)(32,51,52,41)(38,54,42,56)(46,55,50,53)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=g,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6468871);