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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1234 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)(3,4)(5,6)(7,8)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(43,49)(46,50)(51,55)(53,56);(1,3)(2,4)(5,7)(6,8)(12,19)(15,17)(20,24)(22,23)(26,30)(28,36)(31,33)(34,40)(41,42,45,48)(43,50,49,46)(44,47,52,54)(51,56,55,53);(2,6)(3,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39)(41,42)(43,46)(44,47)(45,48)(49,50)(51,53)(52,54)(55,56);(1,2)(3,8)(4,7)(5,6)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)(41,43)(42,46)(44,51)(45,49)(47,53)(48,50)(52,55)(54,56);(1,4,5,8)(2,7,6,3)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)(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=fgh,c^2=ghi,d^2=fghi,e^2=fh,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, 6038570);