Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(11,16)(12,17)(18,20)(21,22,26,27)(23,29,32,28)(24,33,34,25)(30,31,36,35)(40,47)(43,50)(45,51)(48,52);(7,13)(11,16)(14,19)(18,20)(22,27)(24,34)(28,29)(30,36)(37,38,41,44)(39,42,46,49)(40,43,47,50)(45,48,51,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,32)(22,28,27,29)(24,30,34,36)(25,31,33,35)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,34)(22,25,27,33)(23,30,32,36)(28,31,29,35)(37,39,41,46)(38,42,44,49)(40,45,47,51)(43,48,50,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25)(22,24)(23,31)(26,33)(27,34)(28,30)(29,36)(32,35)(37,40)(38,43)(39,45)(41,47)(42,48)(44,50)(46,51)(49,52)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=f,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775872);