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 | |
1242 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)(11,17)(12,19)(15,20)(16,21)(26,30)(27,33)(28,36)(38,40)(41,42)(43,49)(44,52)(45,46)(47,50)(48,53)(51,54)(55,56);(1,2,5,3)(4,6,8,7)(10,14)(15,20)(16,21)(22,24)(26,30)(28,36)(31,34)(35,39)(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)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39)(41,43)(42,47)(44,51)(45,50)(46,49)(48,54)(52,55)(53,56);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40)(41,44,45,53)(42,48,46,52)(43,51,50,56)(47,54,49,55);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)(41,42)(43,47)(44,52)(45,46)(48,53)(49,50)(51,55)(54,56)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=fgh,e^2=fg,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, 5699899);