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)(6,10)(7,13)(12,17)(15,19)(21,22,25,26)(23,29,30,27)(24,28,32,34)(31,36,33,35)(39,45)(40,47)(42,48)(43,50);(8,16)(12,17)(15,19)(18,20)(22,26)(23,30)(24,32)(35,36)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23,25,30)(22,27,26,29)(24,33,32,31)(28,36,34,35)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,34)(33,36)(37,39,41,45)(38,42,44,48)(40,46,47,51)(43,49,50,52);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,24)(22,28)(23,31)(25,32)(26,34)(27,35)(29,36)(30,33)(37,40)(38,43)(39,46)(41,47)(42,49)(44,50)(45,51)(48,52)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,d^2=gh,e^2=fg,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, 5124246);