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 | |
1234 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)(3,8)(4,7)(5,6)(12,19)(16,21)(17,23)(22,24)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39)(44,51)(45,53)(47,49)(48,50);(2,5)(3,4)(10,14)(15,20)(16,21)(22,24)(27,34)(28,37)(30,33)(31,36)(41,42,46,43)(44,49,51,47)(45,48,53,50)(52,55,56,54);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40)(41,43,46,42)(44,49,51,47)(45,50,53,48)(52,55,56,54);(1,4,6,3)(2,7,5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40)(41,44)(42,47)(43,49)(45,52)(46,51)(48,54)(50,55)(53,56);(1,5,6,2)(3,7,4,8)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)(41,45)(42,48)(43,50)(44,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=gi,d^2=h,e^2=f,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, 6178680);