Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,35)(27,32,38,47)(28,39,42,53)(29,34,30,44)(33,40,49,41)(36,50,52,55)(37,46,43,48)(45,54,56,51);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(25,27)(26,32)(28,36)(29,43)(30,37)(31,38)(33,45)(34,48)(35,47)(39,50)(40,54)(41,51)(42,52)(44,46)(49,56)(53,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,29,40,31,42,30,41)(26,33,44,53,35,49,34,39)(27,36,43,54,38,52,37,51)(32,45,46,55,47,56,48,50);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(25,29,31,30)(26,34,35,44)(27,37,38,43)(28,40,42,41)(32,46,47,48)(33,39,49,53)(36,51,52,54)(45,55,56,50);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(27,38)(32,47)(36,52)(37,43)(45,56)(46,48)(50,55)(51,54)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819387);