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 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(4,8)(6,7)(12,19)(16,21)(17,23)(22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39)(41,42)(43,49)(44,50)(45,48)(46,47)(51,54)(52,56)(53,55);(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)(43,50)(45,53)(52,54);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40)(41,43)(42,44)(45,51)(46,50)(47,49)(48,52)(53,56)(54,55);(1,4)(2,6)(3,7)(5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39)(41,44)(42,43)(45,54)(46,49)(47,50)(48,56)(51,55)(52,53);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)(41,45,46,53)(42,48,47,55)(43,51,50,56)(44,52,49,54)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=hi,d^2=fh,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, 5667790);