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)(5,6,9,10)(7,13,15,11)(8,12,17,16)(14,19,20,18)(21,22)(23,29)(24,32)(25,28)(26,31)(27,34)(30,35)(33,36)(37,38,42,45)(39,40,47,43)(41,49,51,44)(46,52,50,48);(6,10)(11,13)(12,16)(18,19)(21,22,25,28)(23,26,31,29)(24,27,34,32)(30,35,33,36)(37,39,42,47)(38,43,45,40)(41,46,51,50)(44,52,49,48);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,35)(32,34)(33,36)(37,40,42,43)(38,39,45,47)(41,48,51,52)(44,46,49,50);(1,2)(3,4)(5,6)(7,11)(8,16)(9,10)(12,17)(13,15)(14,19)(18,20)(21,23,25,31)(22,26,28,29)(24,33,34,30)(27,36,32,35)(37,39)(38,43)(40,45)(41,50)(42,47)(44,48)(46,51)(49,52);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,24,25,34)(22,27,28,32)(23,30,31,33)(26,35,29,36)(37,41,42,51)(38,44,45,49)(39,46,47,50)(40,48,43,52)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgh,d^2=fg,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, 4770376);