Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394358 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2119 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)(5,6,10,15,11,17,12,16)(7,9,19,26,14,27,20,28)(8,21,24,31,23,32,25,13)(18,33,34,29,30,36,35,22);(5,7)(6,9)(8,18,23,30)(10,19)(11,14)(12,20)(13,29,31,22)(15,26)(16,28)(17,27)(21,36,32,33)(24,34,25,35);(2,4)(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,24)(14,30)(15,32)(16,21)(17,31)(19,35)(20,34)(26,36)(27,29)(28,33);(5,9,11,27)(6,14,17,7)(8,22,23,29)(10,28,12,26)(13,30,31,18)(15,19,16,20)(21,34,32,35)(24,36,25,33)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302817);