Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,40)(37,52)(38,53)(39,46)(41,56)(42,49)(43,51)(44,50)(47,54)(48,55);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(27,37)(32,46)(36,52)(38,41)(39,40)(47,48)(53,56)(54,55);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,27,30,40,31,37,29,39)(26,32,34,36,35,46,45,52)(28,41,42,55,43,38,44,54)(33,48,50,56,51,47,49,53);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,31,43)(26,33,35,51)(27,38,37,41)(29,44,30,42)(32,47,46,48)(34,50,45,49)(36,53,52,56)(39,55,40,54);(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,34,35,45)(27,39,37,40)(28,42,43,44)(32,36,46,52)(33,49,51,50)(38,54,41,55)(47,56,48,53)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6831640);