Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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 | |
6262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,4,3)(5,6)(7,8)(9,17)(10,13)(11,15)(12,18)(14,19)(16,20)(21,22)(23,28)(24,32)(25,30)(26,27)(29,34)(31,33)(35,36);(1,2,4,3)(22,27)(23,30)(29,32)(31,36);(5,7)(6,11)(8,13)(9,14)(10,15)(12,19)(16,17)(18,20)(21,23,26,30)(22,28,27,25)(24,31,34,36)(29,35,32,33);(5,8,10,11)(6,7,13,15)(9,16,18,19)(12,14,17,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,33,28,35);(1,3,4,2)(5,7)(6,8)(9,14)(10,15)(11,13)(12,16)(17,19)(18,20)(21,25,26,28)(22,30,27,23)(24,33,34,35)(29,36,32,31);(5,9,10,18)(6,12,13,17)(7,14,15,20)(8,16,11,19)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534347);