Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,6,5)(3,8,4,7)(11,17)(15,20)(18,23)(22,24)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39)(41,42,46,44)(43,49,50,47)(45,52,53,48)(51,54,56,55);(1,3,6,4)(2,7,5,8)(10,14)(11,17)(16,21)(18,23)(28,35)(31,33)(36,40)(38,39)(42,44)(43,50)(45,53)(54,55);(1,2,6,5)(3,7,4,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38)(41,43,46,50)(42,47,44,49)(45,51,53,56)(48,54,52,55);(1,4,6,3)(2,7,5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,44,46,42)(43,49,50,47)(45,52,53,48)(51,55,56,54);(1,5,6,2)(3,7,4,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,45)(42,48)(43,51)(44,52)(46,53)(47,54)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=fi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5941826);