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)(11,17)(12,19)(15,20)(16,21)(25,26)(27,33)(28,34)(29,32)(30,31)(35,38)(36,40)(37,39)(41,42,45,48)(43,49,51,46)(44,52,53,47)(50,54,56,55);(10,14)(15,20)(16,21)(22,24)(26,31)(27,34)(29,37)(36,38)(41,42,45,48)(43,46,51,49)(44,52,53,47)(50,55,56,54);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39)(41,42)(43,49)(44,47)(45,48)(46,51)(50,55)(52,53)(54,56);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,27)(29,38)(30,33)(31,34)(32,40)(35,39)(36,37)(41,43)(42,46)(44,50)(45,51)(47,54)(48,49)(52,55)(53,56);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,37)(26,32,31,39)(27,35,34,40)(28,36,33,38)(41,44,45,53)(42,47,48,52)(43,50,51,56)(46,54,49,55)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=hi,d^2=gh,e^2=g,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, 5636358);