Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(7,13)(8,15)(18,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36)(38,40)(43,45)(44,48)(51,52);(6,10)(7,13)(12,17)(14,19)(22,23)(24,31)(28,30)(32,36)(37,38,42,40)(39,45,47,43)(41,44,50,48)(46,52,49,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35)(37,39,42,47)(38,43,40,45)(41,49,50,46)(44,52,48,51);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35)(37,40,42,38)(39,45,47,43)(41,48,50,44)(46,52,49,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)(37,41,42,50)(38,44,40,48)(39,46,47,49)(43,51,45,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4515605);