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)(8,15)(12,17)(13,19)(18,20)(21,22)(23,27)(24,31)(25,26)(28,33)(29,36)(30,34)(32,35)(38,42)(40,48)(43,45)(46,49);(6,10)(11,16)(12,17)(18,20)(22,26)(27,34)(28,31)(35,36)(37,38,41,42)(39,45,47,43)(40,44,48,50)(46,52,49,51);(5,6,9,10)(7,11,14,16)(8,12,15,17)(13,18,19,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,34)(31,33)(32,36)(37,38)(39,43)(40,44)(41,42)(45,47)(46,51)(48,50)(49,52);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,23,25,30)(22,27,26,34)(24,32,33,29)(28,36,31,35)(37,39,41,47)(38,43,42,45)(40,49,48,46)(44,52,50,51);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,24)(22,28)(23,29)(25,33)(26,31)(27,35)(30,32)(34,36)(37,40,41,48)(38,44,42,50)(39,46,47,49)(43,51,45,52)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fh,d^2=h,e^2=g,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, 4776012);