Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1398 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,50)(38,46)(40,51)(41,47)(42,48)(43,49)(45,53)(52,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(28,41)(33,47)(37,52)(42,43)(45,55)(48,49)(50,56)(53,54);(3,8)(5,7)(12,21)(15,20)(18,24)(22,23)(25,27,30,39,31,40,29,38)(26,32,34,36,35,46,44,51)(28,37,42,53,41,52,43,54)(33,45,49,50,47,55,48,56);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,33)(27,37)(29,43)(30,42)(31,41)(32,45)(34,49)(35,47)(36,50)(38,54)(39,53)(40,52)(44,48)(46,55)(51,56);(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,42,41,43)(32,36,46,51)(33,48,47,49)(37,53,52,54)(45,56,55,50)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855766);