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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 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)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,31,36,44)(28,38,40,53)(29,42,39,33)(32,46,47,41)(35,48,50,56)(37,52,49,45)(43,54,55,51);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,31)(28,35)(29,37)(30,36)(32,43)(33,45)(34,44)(38,48)(39,49)(40,50)(41,51)(42,52)(46,54)(47,55)(53,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,39,46,30,40,29,41)(26,32,42,53,34,47,33,38)(27,35,49,54,36,50,37,51)(31,43,52,56,44,55,45,48);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(27,36)(31,44)(35,50)(37,49)(43,55)(45,52)(48,56)(51,54);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,36)(28,40)(31,44)(32,47)(37,49)(38,53)(41,46)(45,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=f,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, 6820473);