Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(7,13)(8,15)(11,16)(12,17)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36)(37,38,41,44)(39,45,47,42)(40,43,48,50)(46,51,52,49);(6,10)(11,16)(12,17)(18,20)(22,26)(23,30)(28,32)(31,36)(40,48)(43,50)(46,52)(49,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,34)(35,36)(37,39,41,47)(38,42,44,45)(40,46,48,52)(43,49,50,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,50)(49,52);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)(37,40)(38,43)(39,46)(41,48)(42,49)(44,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=g,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5124999);