Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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:
(3,7)(5,8)(11,17)(12,19)(15,20)(16,21)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37)(42,46)(43,49)(44,52)(54,56);(10,14)(15,20)(16,21)(22,24)(27,34)(30,33)(35,40)(37,39)(41,42,45,46)(43,50,49,47)(44,53,52,48)(51,54,55,56);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40)(41,42)(43,50)(44,48)(45,46)(47,49)(51,56)(52,53)(54,55);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39)(41,43)(42,47)(44,51)(45,49)(46,50)(48,54)(52,55)(53,56);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39)(41,44,45,52)(42,48,46,53)(43,51,49,55)(47,54,50,56)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=gi,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4394503);