Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1350 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,31,35)(27,36,40,51)(28,33,41,47)(29,34,30,44)(32,38,46,39)(37,50,52,56)(42,48,43,49)(45,54,55,53);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(28,41)(33,47)(37,52)(42,43)(45,55)(48,49)(50,56)(53,54);(2,6)(3,7)(25,27,29,38,31,40,30,39)(26,32,44,51,35,46,34,36)(28,37,43,54,41,52,42,53)(33,45,48,56,47,55,49,50);(10,15)(14,17)(16,23)(22,24)(25,28,31,41)(26,33,35,47)(27,37,40,52)(29,43,30,42)(32,45,46,55)(34,49,44,48)(36,50,51,56)(38,54,39,53);(3,7)(5,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,42,41,43)(32,36,46,51)(33,48,47,49)(37,53,52,54)(45,56,55,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,d^2=i,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6672119);