Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6910 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)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(2,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(6,11)(10,16)(12,18)(17,20)(23,29)(26,33)(30,35)(34,36);(6,11)(7,13)(12,18)(14,19)(23,29)(26,33)(30,35)(34,36);(6,11)(10,16)(12,18)(17,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^b=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541140);