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)(11,17)(12,19)(22,24)(25,26)(27,30)(28,31)(29,34)(32,38)(33,35)(36,40)(37,39)(41,42,46,47)(43,44,51,48)(45,49,54,55)(50,52,56,53);(1,2,5,3)(4,6,8,7)(10,14)(11,17)(16,21)(18,23)(26,27)(28,35)(32,34)(36,37)(42,47)(44,48)(49,55)(52,53);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,33,35,31)(29,34,38,32)(36,40,37,39)(41,43)(42,48)(44,47)(45,50)(46,51)(49,53)(52,55)(54,56);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,37)(30,35)(32,40)(34,39)(36,38)(41,44,46,48)(42,43,47,51)(45,53,54,52)(49,56,55,50);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,38)(26,32,27,34)(28,36,35,37)(31,39,33,40)(41,45,46,54)(42,49,47,55)(43,50,51,56)(44,52,48,53)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,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^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4796495);