Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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:
(1,2,4,6)(3,7,8,5)(10,14)(11,17)(16,21)(19,23)(25,26)(27,30)(28,31)(29,34)(32,37)(33,35)(36,39)(38,40)(41,42)(43,46)(44,52)(45,48)(47,53)(49,51)(50,55)(54,56);(3,8)(5,7)(10,14)(11,17)(12,20)(22,24)(26,27)(28,35)(29,37)(38,39)(41,42,45,48)(43,49,51,46)(44,47,53,52)(50,55,56,54);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39)(41,42)(43,49)(44,52)(45,48)(46,51)(47,53)(50,54)(55,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,43,45,51)(42,46,48,49)(44,50,53,56)(47,54,52,55);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,44)(42,47)(43,50)(45,53)(46,54)(48,52)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=gh,c^2=fgh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6097641);