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 | |
1256 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)(12,19)(15,20)(18,23)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37)(41,42,45,46)(43,49,50,47)(44,48,53,52)(51,55,56,54);(1,2,5,3)(4,6,8,7)(10,14)(11,17)(16,21)(18,23)(27,34)(30,33)(35,40)(37,39)(42,46)(43,50)(48,52)(51,56);(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)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40)(41,42)(43,47)(44,52)(45,46)(48,53)(49,50)(51,55)(54,56);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39)(41,43,45,50)(42,47,46,49)(44,51,53,56)(48,54,52,55);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39)(41,44,45,53)(42,48,46,52)(43,51,50,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,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, 4637391);