Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,7,10,15)(8,19,21,17,23,30,22,13)(9,24,27,14,28,32,26,18)(20,33,25,34,36,29,35,31);(1,3,4,2)(5,7,12,6)(8,19)(9,18,28,14)(10,15,11,16)(13,21)(17,22)(20,34)(23,30)(24,27,32,26)(25,33)(29,35)(31,36);(2,3)(5,8,10,21,12,23,11,22)(6,13,15,19,7,17,16,30)(9,25,26,36,28,35,27,20)(14,31,24,33,18,34,32,29);(5,9,12,28)(6,14,7,18)(8,20,23,36)(10,27,11,26)(13,29,17,33)(15,32,16,24)(19,34,30,31)(21,35,22,25)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364079);