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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,10,7)(8,13,15,11)(9,12,17,14)(16,19,20,18)(21,22)(23,29)(24,26)(25,28)(27,30)(31,36)(32,33)(34,35)(38,39)(40,47)(41,49)(50,51);(6,7)(8,15)(12,14)(16,20)(22,24)(23,30)(25,33)(35,36)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23)(22,27)(24,29)(25,34)(26,30)(28,36)(31,33)(32,35)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,36,34,35)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,25,26,33)(22,28,24,32)(23,31,30,34)(27,35,29,36)(37,41,42,49)(38,44,39,46)(40,48,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=gh,d^2=i,e^2=g,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, 5188094);