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 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,3)(4,7,8,6)(10,14)(15,20)(16,21)(22,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(41,42,46,49)(43,44,50,47)(45,53,54,48)(51,55,56,52);(1,2,5,3)(4,6,8,7)(11,17)(15,20)(18,23)(22,24)(26,30)(31,33)(32,36)(38,39)(43,50)(44,47)(51,56)(52,55);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)(41,43)(42,47)(44,49)(45,51)(46,50)(48,55)(52,53)(54,56);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40)(41,44,46,47)(42,43,49,50)(45,52,54,55)(48,51,53,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)(41,45,46,54)(42,48,49,53)(43,51,50,56)(44,52,47,55)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fgi,d^2=h,e^2=g,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, 4796456);