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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,21,15)(12,16,22,13)(19,23,24,20)(25,26)(27,36)(28,40)(29,35)(30,32)(31,46)(33,38)(34,41)(37,54)(39,49)(42,53)(43,52)(44,47)(45,51)(48,55)(50,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,22,24)(13,23,16,20)(26,32)(28,41)(33,47)(35,46)(36,49)(37,42)(43,45)(55,56);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,17)(14,22)(15,20)(18,23)(21,24)(25,27,31,38,30,39,29,44)(26,33,35,36,32,47,46,49)(28,42,43,55,41,37,45,56)(34,50,52,54,40,48,51,53);(3,8)(5,7)(10,17)(12,22)(15,18)(19,24)(25,28)(26,34)(27,37)(29,45)(30,41)(31,43)(32,40)(33,48)(35,52)(36,53)(38,56)(39,42)(44,55)(46,51)(47,50)(49,54);(2,6)(5,7)(9,13,14,16)(10,12,17,22)(11,20,21,23)(15,24,18,19)(25,29,30,31)(26,35,32,46)(27,38,39,44)(28,43,41,45)(33,49,47,36)(34,51,40,52)(37,55,42,56)(48,53,50,54)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=g,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, 6821021);