Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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:
(4,8)(6,7)(10,14)(15,20)(16,21)(22,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40)(41,42,45,46)(43,49,51,47)(44,52,54,48)(50,55,53,56);(1,2,5,3)(4,6,8,7)(11,17)(15,20)(18,23)(22,24)(26,30)(31,37)(32,35)(38,39)(42,46)(47,49)(48,52)(55,56);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40)(41,43)(42,47)(44,53)(45,51)(46,49)(48,56)(50,54)(52,55);(1,4,5,8)(2,6,3,7)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39)(41,42)(43,47)(44,48)(45,46)(49,51)(50,55)(52,54)(53,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39)(41,44)(42,48)(43,50)(45,54)(46,52)(47,55)(49,56)(51,53)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,c^2=gi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637295);