Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1162 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,33)(27,35,38,31)(28,32,42,48)(30,34,37,44)(36,49,50,43)(39,47,41,45)(40,46,52,55)(51,56,53,54);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(27,36)(28,40)(30,37)(31,43)(32,46)(34,44)(35,49)(38,50)(39,41)(42,52)(45,47)(48,55);(2,6)(7,8)(25,27,37,50,29,38,30,36)(26,31,44,49,33,35,34,43)(28,41,52,51,42,39,40,53)(32,47,55,54,48,45,46,56);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(27,38)(31,35)(36,50)(39,41)(43,49)(45,47)(51,53)(54,56);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,37,52,29,42,30,40)(26,32,44,55,33,48,34,46)(27,39,50,53,38,41,36,51)(31,45,49,56,35,47,43,54)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=gi,e^2=fgh,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750727);