Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,13,15)(11,14,17,21)(12,16,19,22)(18,20,23,24)(25,26,31,35)(27,36,38,51)(28,41,43,33)(29,34,30,45)(32,39,47,40)(37,52,53,50)(42,49,44,48)(46,54,56,55);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,39,31,38,30,40)(26,32,45,51,35,47,34,36)(28,37,44,55,43,53,42,54)(33,46,48,52,41,56,49,50);(2,5)(6,8)(10,15)(11,17)(16,22)(18,23)(25,28,31,43)(26,33,35,41)(27,37,38,53)(29,44,30,42)(32,46,47,56)(34,49,45,48)(36,50,51,52)(39,55,40,54);(1,3,4,7)(2,6,5,8)(12,19)(16,22)(18,23)(20,24)(27,38)(32,47)(36,51)(37,53)(39,40)(46,56)(50,52)(54,55);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,29,31,30)(26,34,35,45)(27,39,38,40)(28,42,43,44)(32,36,47,51)(33,48,41,49)(37,54,53,55)(46,52,56,50)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=ghi,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6454892);