Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,17,29,15)(7,18,21,12,22,32,35,31)(8,9,25,27,24,13,14,28)(19,26,23,30,34,33,36,20);(5,7,10,21,11,22,29,35)(6,12,15,18,17,31,16,32)(8,23,14,19,24,36,25,34)(9,26,27,30,13,33,28,20);(5,8,10,25,11,24,29,14)(6,13,15,27,17,9,16,28)(7,19,35,36,22,34,21,23)(12,30,32,33,31,20,18,26);(2,4)(5,9,11,13)(6,14,17,25)(7,20,22,30)(8,16,24,15)(10,28,29,27)(12,23,31,36)(18,34,32,19)(21,33,35,26)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=ef,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341214);