Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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:
(1,2,5,3)(4,7,8,6)(10,14)(11,17)(16,21)(19,23)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38)(41,42,46,47)(43,44,48,51)(45,49,54,55)(50,52,53,56);(1,3,5,2)(4,7,8,6)(12,20)(16,21)(19,23)(22,24)(27,33)(30,37)(34,39)(38,40)(42,47)(44,51)(49,55)(52,56);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40)(41,43)(42,44)(45,53)(46,48)(47,51)(49,56)(50,54)(52,55);(1,4,5,8)(2,6,3,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40)(41,44)(42,48)(43,47)(45,52)(46,51)(49,53)(50,55)(54,56);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)(41,45)(42,49)(43,50)(44,52)(46,54)(47,55)(48,53)(51,56)Polycyclic presentation:
2^9|a^2=fgi,b^2=ghi,c^2=gh,d^2=hi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5030436);