Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1254 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)(6,8)(11,17)(12,19)(14,20)(15,22)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40)(41,42,45,48)(43,46,51,49)(44,52,53,47)(50,55,56,54);(2,5)(3,7)(11,17)(14,20)(18,23)(21,24)(26,30)(28,37)(31,33)(34,40)(41,42,45,48)(43,49,51,46)(44,47,53,52)(50,55,56,54);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(41,42)(43,49)(44,47)(45,48)(46,51)(50,55)(52,53)(54,56);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39)(41,43,45,51)(42,46,48,49)(44,50,53,56)(47,54,52,55);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)(41,44)(42,47)(43,50)(45,53)(46,54)(48,52)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5927007);