Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,33,29,35,30,36)(27,37,45,49,39,32,40,47)(28,41,46,52,43,34,44,50)(38,48,54,55,42,51,53,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,39)(26,32,35,37)(28,42,43,38)(30,40,31,45)(33,47,36,49)(34,51,41,48)(44,54,46,53)(50,56,52,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(30,31)(32,47)(34,50)(35,36)(37,49)(40,45)(41,52)(44,46)(48,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28,29,43)(26,34,35,41)(27,38,39,42)(30,44,31,46)(32,48,37,51)(33,50,36,52)(40,53,45,54)(47,55,49,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(26,35)(27,39)(28,43)(33,36)(40,45)(44,46)(48,51)(55,56)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6775811);