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) = C25×C4 |
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 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,17)(12,20)(22,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38)(42,46)(47,49)(48,52)(54,55);(3,8)(5,7)(26,28)(29,37)(31,33)(34,40)(41,42,45,46)(43,49,51,47)(44,52,53,48)(50,54,56,55);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)(41,43,45,51)(42,47,46,49)(44,50,53,56)(48,54,52,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38)(41,42)(43,49)(44,48)(45,46)(47,51)(50,55)(52,53)(54,56);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,44,45,53)(42,48,46,52)(43,50,51,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=fh,e^2=fg,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, 5908875);