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 | |
1611 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)(23,27)(24,26)(25,28)(29,31)(30,35)(32,34)(33,36)(38,39)(40,47)(41,49)(50,51);(6,10)(11,16)(12,17)(18,20)(22,24)(23,29)(28,32)(30,33)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,29)(22,27,24,31)(25,33,34,30)(28,36,32,35)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,22)(23,27,29,31)(25,32,34,28)(30,35,33,36)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,34)(22,28,24,32)(23,30,29,33)(27,35,31,36)(37,41,42,49)(38,44,39,46)(40,48,47,52)(43,50,45,51)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=fghi,e^2=gh,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, 5191335);