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 | |
1179 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)(27,31)(28,33)(29,32)(30,34)(35,41)(36,44)(37,43)(38,45)(39,48)(40,46)(42,50)(47,52)(49,54)(51,55)(53,56);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,36)(26,31,34,44)(28,37,39,49)(29,38,41,50)(32,45,35,42)(33,43,48,54)(40,51,52,56)(46,55,47,53);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,32)(27,36)(29,41)(31,42)(33,46)(34,35)(37,49)(40,52)(43,53)(44,45)(47,48)(54,55);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,39)(26,33,34,48)(27,37,36,49)(29,40,41,52)(31,43,44,54)(32,46,35,47)(38,51,50,56)(42,53,45,55);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(26,34)(27,36)(28,39)(32,35)(38,50)(40,52)(43,54)(53,55)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=hi,d^2=h,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=dfhi,e^b=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, 6559796);