Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,28,11,17,12,16)(7,18,20,13,21,8,14,29)(9,23,25,15,27,32,26,31)(19,30,24,33,34,36,35,22);(5,7,10,20,11,21,12,14)(6,13,16,18,17,29,28,8)(9,24,25,34,27,35,26,19)(15,22,23,36,31,33,32,30);(5,8,12,29,11,18,10,13)(6,14,16,7,17,20,28,21)(9,22,25,36,27,33,26,30)(15,24,32,19,31,35,23,34);(2,4)(5,9,11,27)(6,15,17,31)(7,19,21,34)(8,22,18,33)(10,26,12,25)(13,30,29,36)(14,24,20,35)(16,32,28,23)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341272);