Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | |
1426 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)(27,32,30,45)(28,38,41,47)(29,34,44,52)(33,36,49,39)(37,46,42,50)(40,55,43,56)(48,53,51,54);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(25,27,31,30)(26,32,35,45)(28,39,41,36)(29,42,44,37)(33,47,49,38)(34,50,52,46)(40,53,43,54)(48,55,51,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,27,36,31,41,30,39)(26,33,45,47,35,49,32,38)(29,43,37,54,44,40,42,53)(34,51,50,55,52,48,46,56);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(25,29)(26,34)(27,37)(28,40)(30,42)(31,44)(32,46)(33,48)(35,52)(36,53)(38,55)(39,54)(41,43)(45,50)(47,56)(49,51);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(28,41)(29,44)(33,49)(34,52)(36,39)(37,42)(38,47)(46,50)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,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=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819489);