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)(11,17)(12,19)(22,24)(25,26,29,30)(27,31,33,38)(28,35,37,32)(34,40,36,39)(41,42)(43,49)(44,46)(45,48)(47,51)(50,55)(52,53)(54,56);(1,2,5,3)(4,6,8,7)(10,14)(11,17)(16,21)(18,23)(26,30)(27,33)(32,35)(34,36)(42,44)(47,49)(48,52)(54,55);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)(41,43,46,51)(42,47,44,49)(45,50,53,56)(48,54,52,55);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39)(41,44,46,42)(43,49,51,47)(45,48,53,52)(50,54,56,55);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40)(41,45)(42,48)(43,50)(44,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,c^2=fgi,d^2=fh,e^2=fg,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, 4796514);