Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1240 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)(10,14)(12,19)(15,20)(18,23)(25,26,29,30)(27,33,34,31)(28,36,38,32)(35,39,37,40)(41,42,46,44)(43,47,49,50)(45,52,53,48)(51,56,55,54);(1,2,5,4)(3,6,7,8)(10,14)(11,17)(16,21)(18,23)(26,30)(27,34)(32,36)(35,37)(43,49)(47,50)(51,55)(54,56);(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40)(41,43)(42,47)(44,50)(45,51)(46,49)(48,54)(52,56)(53,55);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,34)(26,31,30,33)(28,37,38,35)(32,40,36,39)(41,44,46,42)(43,50,49,47)(45,52,53,48)(51,56,55,54);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,38)(26,32,30,36)(27,35,34,37)(31,39,33,40)(41,45)(42,48)(43,51)(44,52)(46,53)(47,54)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=fg,d^2=fgh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4825884);