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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1670 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)(7,10)(8,13)(9,14)(11,15)(12,17)(16,18)(19,20)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35)(38,39)(40,47)(44,46)(48,52);(6,7)(8,15)(9,17)(18,19)(23,30)(24,33)(26,29)(27,32)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,7,10,6)(8,13,15,11)(9,12,17,14)(16,18,20,19)(21,23)(22,26)(24,31)(25,30)(27,34)(28,29)(32,35)(33,36)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,22)(23,29)(24,27)(25,28)(26,30)(31,35)(32,33)(34,36)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,24)(22,27)(23,31)(25,33)(26,34)(28,32)(29,35)(30,36)(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=f,d^2=hi,e^2=gh,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, 4992008);