Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1562 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)(5,6)(7,13)(8,10)(9,12)(11,15)(14,19)(16,17)(18,20)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35)(37,38,42,40)(39,45,47,43)(41,44,50,48)(46,52,49,51);(6,8)(11,13)(12,16)(18,19)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34)(37,39)(38,43)(40,45)(41,46)(42,47)(44,51)(48,52)(49,50);(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36)(37,40,42,38)(39,45,47,43)(41,48,50,44)(46,52,49,51);(1,2)(3,4)(5,8,10,6)(7,13,15,11)(9,12,17,16)(14,18,20,19)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35)(37,39)(38,43)(40,45)(41,49)(42,47)(44,52)(46,50)(48,51);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(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=fg,b^2=h,c^2=fgh,d^2=g,e^2=f,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, 4770369);