Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1178 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26,30,34)(27,31,38,47)(28,32,40,49)(29,33,43,52)(35,44,53,55)(36,45,37,46)(39,48,54,56)(41,50,42,51);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,38)(26,31,34,47)(28,39,40,54)(29,36,43,37)(32,48,49,56)(33,45,52,46)(35,42,53,41)(44,51,55,50);(2,5)(6,8)(25,28,30,40)(26,32,34,49)(27,35,38,53)(29,42,43,41)(31,44,47,55)(33,51,52,50)(36,39,37,54)(45,48,46,56);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(28,40)(32,49)(35,53)(39,54)(41,42)(44,55)(48,56)(50,51);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,36)(28,40)(29,43)(31,45)(32,49)(33,52)(35,54)(37,38)(39,53)(44,56)(46,47)(48,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6612028);