Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(12,20)(15,21)(19,24)(22,23)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,38)(37,51)(39,46)(40,48)(41,50)(42,47)(43,49)(45,53)(52,55)(54,56);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27,30,38,31,39,29,42)(26,32,34,36,35,47,44,46)(28,37,40,53,41,54,43,52)(33,45,49,51,50,55,48,56);(1,4,5,2)(3,6,8,7)(11,18)(14,17)(19,24)(22,23)(25,28,31,41)(26,33,35,50)(27,37,39,54)(29,43,30,40)(32,45,47,55)(34,49,44,48)(36,51,46,56)(38,53,42,52);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,34,35,44)(27,38,39,42)(28,40,41,43)(32,46,47,36)(33,48,50,49)(37,52,54,53)(45,51,55,56)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fg,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7094385);