Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1363 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,30,36,31,37,29,35)(27,32,40,49,41,50,45,53)(28,42,43,52,44,34,46,33)(38,48,55,51,39,56,54,47);(1,3,4,8)(2,5,6,7)(10,14)(15,20)(16,21)(22,24)(25,27,31,41)(26,32,37,50)(28,39,44,38)(29,45,30,40)(33,51,52,47)(34,48,42,56)(35,53,36,49)(43,54,46,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,31,44)(26,33,37,52)(27,38,41,39)(29,43,30,46)(32,47,50,51)(34,35,42,36)(40,54,45,55)(48,49,56,53);(2,6)(5,7)(11,17)(15,20)(18,23)(22,24)(25,28)(26,34)(27,39)(29,46)(30,43)(31,44)(32,48)(33,36)(35,52)(37,42)(38,41)(40,54)(45,55)(47,53)(49,51)(50,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,31,30)(26,35,37,36)(27,40,41,45)(28,43,44,46)(32,49,50,53)(33,42,52,34)(38,54,39,55)(47,56,51,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,e^b=ei,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539078);