Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | |
1128 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)(25,26,29,31)(27,35,38,32)(28,39,41,33)(30,34,43,44)(36,45,51,47)(37,46,52,49)(40,48,42,53)(50,56,54,55);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,31)(27,36)(28,40)(30,43)(32,45)(33,48)(35,47)(38,51)(39,53)(41,42)(46,49)(50,54);(2,5)(6,8)(25,27,30,36,29,38,43,51)(26,32,34,47,31,35,44,45)(28,37,42,50,41,52,40,54)(33,46,48,56,39,49,53,55);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(26,31)(28,41)(32,35)(34,44)(37,52)(40,42)(45,47)(50,54);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,30,42,29,41,43,40)(26,33,34,48,31,39,44,53)(27,37,36,50,38,52,51,54)(32,46,47,56,35,49,45,55)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6733564);