Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(11,17)(12,19)(15,20)(16,21)(25,26,29,30)(27,33,34,31)(28,32,36,37)(35,39,40,38)(41,42)(43,47)(44,46)(45,52)(48,51)(49,54)(50,56)(53,55);(10,14)(15,20)(16,21)(22,24)(26,30)(27,34)(28,36)(38,39)(41,43,46,51)(42,47,44,48)(45,53,54,50)(49,56,52,55);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,37)(29,30)(32,36)(33,34)(35,39)(38,40)(41,43)(42,48)(44,47)(45,50)(46,51)(49,56)(52,55)(53,54);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,34)(26,31,30,33)(28,35,36,40)(32,38,37,39)(41,44,46,42)(43,48,51,47)(45,52,54,49)(50,56,53,55);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)(41,45)(42,49)(43,50)(44,52)(46,54)(47,55)(48,56)(51,53)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=hi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5246622);