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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1558 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)(8,15)(11,16)(14,19)(21,22)(23,27)(24,31)(25,26)(28,33)(29,36)(30,34)(32,35)(37,38,42,45)(39,43,40,46)(41,44,49,51)(47,50,48,52);(6,10)(7,13)(12,17)(14,19)(22,26)(27,34)(28,31)(35,36)(37,39,42,40)(38,43,45,46)(41,47,49,48)(44,50,51,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,34)(31,33)(32,36)(37,40,42,39)(38,43,45,46)(41,48,49,47)(44,50,51,52);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,30)(22,27,26,34)(24,32,33,29)(28,36,31,35)(37,38)(39,46)(40,43)(41,44)(42,45)(47,52)(48,50)(49,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,28)(23,29)(25,33)(26,31)(27,35)(30,32)(34,36)(37,41,42,49)(38,44,45,51)(39,47,40,48)(43,50,46,52)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=fgh,d^2=f,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, 4775503);