Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1242 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)(11,17)(12,19)(15,20)(16,21)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37)(41,42,46,47)(43,48,49,44)(45,50,54,55)(51,53,56,52);(1,2,5,3)(4,6,8,7)(10,14)(15,20)(16,21)(22,24)(28,36)(31,38)(34,40)(37,39)(42,47)(43,49)(50,55)(51,56);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40)(41,43)(42,48)(44,47)(45,51)(46,49)(50,53)(52,55)(54,56);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39)(41,44,46,48)(42,49,47,43)(45,53,54,52)(50,51,55,56);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)(41,45)(42,50)(43,51)(44,52)(46,54)(47,55)(48,53)(49,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=fgi,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5699944);