Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,31,35)(27,36,39,47)(28,33,40,49)(29,34,30,45)(32,43,48,38)(37,53,41,56)(42,51,44,52)(46,54,50,55);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(28,40)(33,49)(37,41)(42,44)(46,50)(51,52)(53,56)(54,55);(2,5)(6,8)(25,27,29,43,31,39,30,38)(26,32,45,47,35,48,34,36)(28,41,44,55,40,37,42,54)(33,50,51,53,49,46,52,56);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,28,31,40)(26,33,35,49)(27,37,39,41)(29,44,30,42)(32,46,48,50)(34,52,45,51)(36,53,47,56)(38,55,43,54);(2,5)(6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,31,30)(26,34,35,45)(27,38,39,43)(28,42,40,44)(32,47,48,36)(33,51,49,52)(37,54,41,55)(46,53,50,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6695223);