Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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:
(1,2,6,4)(3,8,5,7)(11,17)(12,19)(14,20)(15,22)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38)(41,42)(43,48)(44,49)(45,50)(46,47)(51,56)(52,53)(54,55);(1,2,6,4)(3,7,5,8)(11,17)(14,20)(18,23)(21,24)(26,30)(27,33)(28,36)(38,40)(42,47)(44,48)(50,55)(52,56);(1,3,6,5)(2,7,4,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39)(41,43)(42,48)(44,47)(45,53)(46,49)(50,52)(51,54)(55,56);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40)(41,44,46,48)(42,49,47,43)(45,52,54,56)(50,53,55,51);(1,5,6,3)(2,7,4,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)(41,45)(42,50)(43,51)(44,52)(46,54)(47,55)(48,56)(49,53)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6159858);