Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,37)(28,41)(29,36)(30,35)(31,33)(32,40)(34,44)(38,48)(39,49)(42,51)(43,52)(45,47)(46,50)(53,56)(54,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,40)(26,32,35,37)(28,42,44,38)(29,45,31,39)(33,47,36,49)(34,51,41,48)(43,53,46,54)(50,56,52,55);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,33)(29,31)(32,47)(34,50)(35,36)(37,49)(39,45)(41,52)(43,46)(48,55)(51,56)(53,54);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,44)(26,34,35,41)(27,38,40,42)(29,46,31,43)(32,48,37,51)(33,50,36,52)(39,54,45,53)(47,55,49,56);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(25,29,30,31)(26,33,35,36)(27,39,40,45)(28,43,44,46)(32,49,37,47)(34,52,41,50)(38,53,42,54)(48,55,51,56)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=h,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6559626);