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 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36)(41,42)(43,50)(44,47)(45,48)(46,49)(51,55)(52,54)(53,56);(2,5)(6,8)(10,11)(13,21)(15,17)(19,24)(41,43,46,44)(42,47,49,50)(45,52,53,51)(48,55,56,54);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36)(41,44,46,43)(42,47,49,50)(45,52,53,51)(48,54,56,55);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40)(41,42)(43,47)(44,50)(45,48)(46,49)(51,54)(52,55)(53,56);(1,3)(2,6)(4,7)(5,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)(41,45,46,53)(42,48,49,56)(43,51,44,52)(47,54,50,55)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fgh,d^2=gh,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, 5894235);