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)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40)(41,42,45,48)(43,46,51,49)(44,52,53,47)(50,55,56,54);(10,14)(11,18)(12,21)(22,23)(26,31)(28,32)(29,37)(34,40)(41,42,45,48)(43,49,51,46)(44,47,53,52)(50,55,56,54);(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27,30,35)(26,32,31,28)(29,34,37,40)(33,39,38,36)(41,42)(43,46)(44,47)(45,48)(49,51)(50,54)(52,53)(55,56);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40)(41,43,45,51)(42,46,48,49)(44,50,53,56)(47,54,52,55);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)(41,44)(42,47)(43,50)(45,53)(46,54)(48,52)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fi,d^2=gi,e^2=fgh,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, 4431914);