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×#3505 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6856 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)(7,9)(8,12)(10,11)(13,17)(14,16)(15,18)(19,20)(21,22)(23,29)(24,28)(25,30)(26,27)(31,36)(32,33)(34,35);(8,15)(12,18)(14,19)(16,20)(21,23)(22,25)(24,32)(26,30)(27,29)(28,36)(31,34)(33,35);(1,3,4,2)(6,11)(7,13)(12,18)(14,19)(22,27)(23,30)(28,35)(31,32);(5,7,10,13)(6,9,11,17)(8,14,15,19)(12,16,18,20)(21,23)(22,25)(24,31)(26,30)(27,29)(28,33)(32,34)(35,36);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,24,26,34)(22,28,27,35)(23,31,30,32)(25,33,29,36);(5,9)(6,13)(7,11)(8,16)(10,17)(12,19)(14,18)(15,20)(21,25)(22,23)(24,33)(26,29)(27,30)(28,31)(32,35)(34,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gh,e^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, 7533598);