Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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:
(2,5)(6,8)(11,17)(12,19)(15,20)(16,21)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40)(41,42,46,48)(43,49,51,44)(45,53,55,47)(50,52,56,54);(10,14)(15,20)(16,21)(22,24)(26,30)(31,33)(32,36)(39,40)(41,43,46,51)(42,44,48,49)(45,50,55,56)(47,52,53,54);(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)(41,43)(42,44)(45,50)(46,51)(47,52)(48,49)(53,54)(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)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39)(41,44)(42,43)(45,54)(46,49)(47,56)(48,51)(50,53)(52,55);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)(41,45,46,55)(42,47,48,53)(43,50,51,56)(44,52,49,54)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4819396);