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)(27,35)(28,33)(29,31)(30,34)(32,39)(36,46)(37,45)(38,50)(40,53)(41,47)(42,44)(43,49)(48,52)(51,55)(54,56);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(26,31)(27,36)(28,40)(30,37)(32,46)(33,49)(35,48)(39,52)(42,43)(44,53)(47,50)(51,54);(2,6)(7,8)(25,27,37,52,29,39,30,36)(26,32,45,46,31,35,34,48)(28,41,40,51,42,38,43,54)(33,50,53,56,44,47,49,55);(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)(28,42)(32,35)(34,45)(38,41)(40,43)(46,48)(51,54);(1,4,5,3)(2,8,6,7)(11,19)(14,17)(18,24)(22,23)(25,28,30,43,29,42,37,40)(26,33,34,49,31,44,45,53)(27,38,36,51,39,41,52,54)(32,47,48,56,35,50,46,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=hi,d^2=fg,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6748814);