Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 52 points:
(1,2,4,3)(5,6,10,11)(7,8,13,16)(9,12,17,18)(14,15,19,20)(22,27)(25,37)(28,31)(29,38)(33,47)(36,48)(40,43)(46,52);(6,11)(7,13)(12,18)(14,19)(24,35)(29,38)(32,45)(36,48)(40,43)(42,49)(46,52)(50,51);(5,7)(6,8)(9,14)(10,13)(11,16)(12,15)(17,19)(18,20)(21,22,26,27)(23,31,34,28)(24,29,35,38)(25,30,37,39)(32,43,45,40)(33,44,47,41)(36,42,48,49)(46,51,52,50);(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(21,23,26,34)(22,28,27,31)(24,32,35,45)(25,33,37,47)(29,40,38,43)(30,41,39,44)(36,46,48,52)(42,50,49,51);(1,3,4,2)(21,24)(22,29)(23,32)(25,36)(26,35)(27,38)(28,40)(30,42)(31,43)(33,46)(34,45)(37,48)(39,49)(41,50)(44,51)(47,52);(5,9,10,17)(6,12,11,18)(7,14,13,19)(8,15,16,20)(21,25)(22,30)(23,33)(24,36)(26,37)(27,39)(28,41)(29,42)(31,44)(32,46)(34,47)(35,48)(38,49)(40,50)(43,51)(45,52)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533712);