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)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38)(42,43)(47,49)(48,50)(54,55);(3,8)(5,7)(10,14)(11,17)(12,20)(22,24)(26,30)(27,33)(28,36)(38,40)(41,42,46,43)(44,49,52,47)(45,50,53,48)(51,54,56,55);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39)(41,43,46,42)(44,49,52,47)(45,48,53,50)(51,54,56,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40)(41,44)(42,47)(43,49)(45,51)(46,52)(48,54)(50,55)(53,56);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)(41,45,46,53)(42,48,43,50)(44,51,52,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=fgh,e^2=gh,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, 6097630);