Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,18,21,15)(12,16,22,13)(19,23,24,20)(25,26,30,35)(27,37,40,32)(28,41,43,50)(29,36,31,34)(33,45,48,42)(38,52,54,56)(39,47,44,49)(46,55,51,53);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,18)(12,19,22,24)(13,23,16,20)(25,27)(26,32)(28,38)(29,44)(30,40)(31,39)(33,46)(34,49)(35,37)(36,47)(41,56)(42,55)(43,54)(45,53)(48,51)(50,52);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,17)(14,22)(15,20)(18,23)(21,24)(25,28,31,42,30,43,29,45)(26,33,36,41,35,48,34,50)(27,38,39,55,40,54,44,53)(32,46,47,56,37,51,49,52);(3,8)(5,7)(10,17)(12,22)(15,18)(19,24)(25,29,30,31)(26,34,35,36)(27,39,40,44)(28,42,43,45)(32,47,37,49)(33,41,48,50)(38,53,54,55)(46,52,51,56);(2,6)(5,7)(9,13,14,16)(10,12,17,22)(11,20,21,23)(15,24,18,19)(28,43)(33,48)(38,54)(41,50)(42,45)(46,51)(52,56)(53,55)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820976);