Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1208 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,45,30,35,29,34)(27,36,38,32,40,47,39,48)(28,41,42,50,44,33,43,49)(37,52,55,56,54,51,53,46);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,27)(26,32)(28,37)(29,38)(30,40)(31,39)(33,46)(34,47)(35,48)(36,45)(41,56)(42,53)(43,55)(44,54)(49,51)(50,52);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28,30,44)(26,33,35,41)(27,37,40,54)(29,43,31,42)(32,46,48,56)(34,50,45,49)(36,51,47,52)(38,55,39,53);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(25,29,30,31)(26,34,35,45)(27,38,40,39)(28,42,44,43)(32,47,48,36)(33,49,41,50)(37,53,54,55)(46,51,56,52)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=f,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855448);