Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| 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 | |
1222 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)(5,6,10,27,11,15,17,16)(7,12,26,30,19,14,9,25)(8,20,28,32,23,33,24,13)(18,34,35,29,21,36,22,31);(5,7,17,26,11,19,10,9)(6,12,27,25,15,14,16,30)(8,21,28,35,23,18,24,22)(13,29,33,36,32,31,20,34);(5,8)(6,13)(7,18)(9,22)(10,28)(11,23)(12,29)(14,31)(15,32)(16,33)(17,24)(19,21)(20,27)(25,34)(26,35)(30,36);(2,4)(5,9,17,19,11,26,10,7)(6,14,16,30,15,12,27,25)(8,22,24,21,23,35,28,18)(13,31,33,36,32,29,20,34)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,d^2=efi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337659);