Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,10)(7,13,14,11)(8,16,17,12)(15,18,20,19)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33)(38,39)(40,47)(44,46)(48,52);(6,10)(7,14)(8,17)(18,19)(24,32)(27,34)(30,36)(33,35)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,6)(7,11)(8,16)(9,10)(12,17)(13,14)(15,19)(18,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2,3,4)(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)(37,41,42,49)(38,44,39,46)(40,48,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=gh,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4989844);