Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,3)(4,7,8,6)(10,14)(15,20)(16,21)(22,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38)(41,42)(43,44)(45,53)(46,47)(48,52)(49,55)(50,54)(51,56);(1,2,5,3)(4,6,8,7)(11,17)(15,20)(18,23)(22,24)(26,28)(31,33)(32,36)(38,39)(42,47)(43,48)(49,53)(50,56);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39)(41,43,46,48)(42,44,47,52)(45,50,55,56)(49,51,53,54);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39)(41,44)(42,48)(43,47)(45,54)(46,52)(49,50)(51,55)(53,56);(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)(41,45,46,55)(42,49,47,53)(43,50,48,56)(44,51,52,54)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=gi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4796477);