Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1185 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)(27,35,37,31)(28,32,40,45)(30,34,41,46)(36,47,49,42)(38,48,51,43)(39,44,52,54)(50,55,56,53);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(27,36)(28,39)(30,41)(31,42)(32,44)(34,46)(35,47)(37,49)(38,51)(40,52)(43,48)(45,54);(2,6)(7,8)(25,27,30,36,29,37,41,49)(26,31,34,42,33,35,46,47)(28,38,39,50,40,51,52,56)(32,43,44,53,45,48,54,55);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(27,37)(31,35)(36,49)(38,51)(42,47)(43,48)(50,56)(53,55);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,30,39,29,40,41,52)(26,32,34,44,33,45,46,54)(27,38,36,50,37,51,49,56)(31,43,42,53,35,48,47,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=gi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750660);