Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1538 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)(6,10)(7,13)(12,17)(14,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38,42,45)(39,46,40,43)(41,44,49,52)(47,51,48,50);(7,13)(11,16)(14,19)(18,20)(23,29)(27,30)(31,35)(34,36)(37,39,42,40)(38,43,45,46)(41,48,49,47)(44,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35)(37,40,42,39)(38,43,45,46)(41,48,49,47)(44,50,52,51);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,38)(39,43)(40,46)(41,44)(42,45)(47,50)(48,51)(49,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)(37,41,42,49)(38,44,45,52)(39,47,40,48)(43,50,46,51)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4375587);