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×#3545 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,25)(24,27)(26,32)(28,30)(29,34)(31,35)(33,36);(5,7,10,16)(6,11,12,18)(8,14,13,19)(9,15,17,20)(21,23)(22,28)(24,25)(26,31)(27,30)(29,35)(32,33)(34,36);(6,12)(8,13)(11,18)(14,19)(22,24)(23,30)(29,32)(31,36);(5,8,10,13)(6,9,12,17)(7,14,16,19)(11,15,18,20)(21,24,27,22)(23,25,30,28)(26,32,34,29)(31,33,36,35);(1,3,4,2)(5,7)(6,11)(8,14)(9,15)(10,16)(12,18)(13,19)(17,20)(21,25)(22,23)(24,30)(26,33)(27,28)(29,31)(32,36)(34,35);(5,9)(6,13)(7,15)(8,12)(10,17)(11,19)(14,18)(16,20)(21,26,27,34)(22,29,24,32)(23,31,30,36)(25,33,28,35)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,e^2=i,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, 7533700);