Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1246 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40)(43,49)(44,51)(46,52)(47,54);(10,11)(15,17)(16,18)(22,23)(26,30)(31,33)(32,36)(39,40)(41,42,45,48)(43,46,49,52)(44,47,51,54)(50,53,55,56);(1,2,4,6)(3,5,7,8)(9,11,14,10)(12,17,20,15)(13,16,21,18)(19,22,24,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39)(41,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,56);(1,3)(2,5)(4,7)(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)(41,43,45,49)(42,46,48,52)(44,50,51,55)(47,53,54,56);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37)(41,44)(42,47)(43,50)(45,51)(46,53)(48,54)(49,55)(52,56)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fghi,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4546363);