Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1258 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)(15,18)(16,21)(22,24)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,39,40,38)(41,42,46,44)(43,47,51,49)(45,52,53,48)(50,55,56,54);(3,8)(5,7)(11,17)(15,18)(19,23)(22,24)(26,30)(28,37)(31,33)(34,40)(41,42,46,44)(43,49,51,47)(45,48,53,52)(50,55,56,54);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)(41,43,46,51)(42,47,44,49)(45,50,53,56)(48,54,52,55);(1,3)(2,5)(4,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(41,44,46,42)(43,49,51,47)(45,52,53,48)(50,55,56,54);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)(41,45)(42,48)(43,50)(44,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5909067);