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 | |
1238 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)(12,20)(22,24)(25,26,30,29)(27,33,34,31)(28,32,36,37)(35,39,40,38)(44,51)(47,54)(49,55)(52,56);(10,14)(12,20)(15,18)(19,23)(26,29)(27,34)(32,37)(35,40)(41,42,45,48)(43,46,50,53)(44,47,51,54)(49,52,55,56);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,34)(26,31,29,33)(28,35,36,40)(32,38,37,39)(41,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,56);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)(41,43,45,50)(42,46,48,53)(44,49,51,55)(47,52,54,56);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,26)(27,33,34,31)(28,32,36,37)(35,38,40,39)(41,44,45,51)(42,47,48,54)(43,49,50,55)(46,52,53,56)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=fi,d^2=fhi,e^2=fgh,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, 5791455);