Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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:
(1,2,5,3)(4,7,8,6)(10,14)(11,17)(16,21)(19,23)(25,26,29,30)(27,31,34,37)(28,32,35,38)(33,39,36,40)(41,42,45,48)(43,46,49,53)(44,51,52,47)(50,56,55,54);(1,2,5,3)(4,6,8,7)(12,20)(16,21)(19,23)(22,24)(26,30)(28,35)(31,37)(33,36)(43,49)(46,53)(50,55)(54,56);(1,3,5,2)(4,7,8,6)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40)(41,43)(42,46)(44,50)(45,49)(47,54)(48,53)(51,56)(52,55);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,34)(26,31,30,37)(28,36,35,33)(32,40,38,39)(41,42)(43,46)(44,47)(45,48)(49,53)(50,54)(51,52)(55,56);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,28)(26,32)(27,33)(29,35)(30,38)(31,39)(34,36)(37,40)(41,44,45,52)(42,47,48,51)(43,50,49,55)(46,54,53,56)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=fghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5860955);