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 | |
1627 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)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36)(38,39)(40,47)(41,49)(50,51);(8,16)(12,17)(15,19)(18,20)(22,26)(23,30)(28,32)(31,36)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,34)(35,36)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2,3,4)(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,19)(17,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)(37,41,42,49)(38,44,39,46)(40,48,47,52)(43,50,45,51)Polycyclic presentation:
2^9|b^2=fh,c^2=fh,d^2=fgi,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, 5188201);