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 | |
1654 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)(5,6,9,10)(7,11,15,13)(8,16,18,12)(14,20,17,19)(21,22,26,29)(23,30,28,25)(24,32,35,27)(31,34,33,36)(38,39)(40,47)(41,49)(50,51);(6,10)(11,13)(12,16)(19,20)(23,28)(25,30)(31,33)(34,36)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,7,9,15)(6,11,10,13)(8,17,18,14)(12,20,16,19)(21,23,26,28)(22,25,29,30)(24,33,35,31)(27,36,32,34)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2,3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,18)(17,19)(21,24,26,35)(22,27,29,32)(23,31,28,33)(25,34,30,36)(37,40)(38,43)(39,45)(41,48)(42,47)(44,50)(46,51)(49,52);(5,8,9,18)(6,12,10,16)(7,14,15,17)(11,19,13,20)(21,25)(22,28)(23,29)(24,34)(26,30)(27,33)(31,32)(35,36)(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=gh,c^2=fh,d^2=fi,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, 5187544);