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 | |
1643 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)(7,13)(8,10)(9,12)(11,14)(15,20)(16,17)(18,19)(21,22,26,27)(23,28,32,36)(24,33,29,25)(30,35,34,31)(38,39)(40,47)(41,49)(50,51);(6,8)(7,14)(9,17)(19,20)(22,27)(24,29)(28,36)(30,34)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,7)(6,11)(8,13)(9,18)(10,14)(12,20)(15,17)(16,19)(21,23,26,32)(22,28,27,36)(24,34,29,30)(25,35,33,31)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2,3,4)(5,8,10,6)(7,13,14,11)(9,16,17,12)(15,20,18,19)(21,24,26,29)(22,25,27,33)(23,30,32,34)(28,31,36,35)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,9,10,17)(6,12,8,16)(7,15,14,18)(11,19,13,20)(21,25)(22,29)(23,31)(24,27)(26,33)(28,34)(30,36)(32,35)(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=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, 5188032);