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)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,29,31)(27,32,38,43)(28,39,41,33)(30,34,36,44)(35,47,50,45)(37,51,52,46)(40,48,42,53)(49,55,54,56);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(26,31)(27,35)(28,40)(30,36)(32,45)(33,48)(38,50)(39,53)(41,42)(43,47)(46,51)(49,54);(2,6)(7,8)(25,27,36,50,29,38,30,35)(26,32,44,45,31,43,34,47)(28,37,40,54,41,52,42,49)(33,46,53,55,39,51,48,56);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,31)(28,41)(32,43)(34,44)(37,52)(40,42)(45,47)(49,54);(1,4,5,3)(2,8,6,7)(11,19)(14,17)(18,24)(22,23)(25,28,30,42,29,41,36,40)(26,33,34,48,31,39,44,53)(27,37,35,49,38,52,50,54)(32,46,47,56,43,51,45,55)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=hi,d^2=fgi,e^2=gh,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, 6748693);