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)(11,17)(15,20)(18,23)(22,24)(25,26)(27,31)(28,30)(29,32)(33,35)(34,38)(36,37)(39,40)(41,42,45,48)(43,49,51,46)(44,47,53,56)(50,55,52,54);(2,5)(6,8)(10,14)(11,17)(16,21)(18,23)(26,28)(27,33)(29,37)(38,40)(41,43,45,51)(42,46,48,49)(44,52,53,50)(47,55,56,54);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,33)(26,31,28,35)(29,34,37,39)(32,38,36,40)(41,43)(42,46)(44,50)(45,51)(47,54)(48,49)(52,53)(55,56);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,26)(27,31,33,35)(29,36,37,32)(34,38,39,40)(41,42)(43,46)(44,47)(45,48)(49,51)(50,54)(52,55)(53,56);(1,3)(2,6)(4,7)(5,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,44,45,53)(42,47,48,56)(43,50,51,52)(46,54,49,55)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=gh,e^2=fgh,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, 5888165);