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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(10,14)(12,20)(15,18)(19,23)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40)(43,49)(46,50)(51,55)(53,56);(2,4)(3,8)(10,14)(11,17)(16,21)(19,23)(26,27)(29,37)(31,33)(35,40)(41,42,45,48)(43,50,49,46)(44,47,52,54)(51,56,55,53);(1,3)(2,6)(4,7)(5,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38)(41,42)(43,46)(44,47)(45,48)(49,50)(51,53)(52,54)(55,56);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)(41,43)(42,46)(44,51)(45,49)(47,53)(48,50)(52,55)(54,56);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)(41,44,45,52)(42,47,48,54)(43,51,49,55)(46,53,50,56)Polycyclic presentation:
2^9|a^2=fghi,b^2=gh,c^2=ghi,d^2=fghi,e^2=gh,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, 6038824);