Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1238 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)(10,14)(11,17)(12,20)(22,24)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38)(41,42,45,48)(43,49,51,46)(44,47,52,54)(50,55,56,53);(1,2,4,6)(3,5,7,8)(10,14)(12,20)(15,18)(19,23)(26,27)(28,35)(32,34)(36,40)(44,52)(47,54)(50,56)(53,55);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,31,35,33)(29,32,37,34)(36,39,40,38)(41,42)(43,49)(44,47)(45,48)(46,51)(50,55)(52,54)(53,56);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,43,45,51)(42,46,48,49)(44,50,52,56)(47,53,54,55);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,44)(42,47)(43,50)(45,52)(46,53)(48,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=f,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5214153);