Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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:
(1,2,4,6)(3,7,8,5)(11,17)(15,20)(18,23)(22,24)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38)(42,46)(44,52)(47,49)(50,56);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(26,27)(28,35)(32,34)(36,40)(41,42,45,46)(43,49,51,47)(44,53,52,48)(50,54,56,55);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,31,35,33)(29,32,37,34)(36,39,40,38)(41,42)(43,47)(44,53)(45,46)(48,52)(49,51)(50,55)(54,56);(1,3)(2,5)(4,8)(6,7)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,43,45,51)(42,47,46,49)(44,50,52,56)(48,54,53,55);(1,2)(3,5)(4,6)(7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,44,45,52)(42,48,46,53)(43,50,51,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=fg,d^2=fgh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6092132);