Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495573 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21408 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60698 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15)(4,19,23,28)(5,10)(6,12)(7,14)(8,13)(9,22,27,25)(11,16)(17,20,29,21)(18,24)(26,31,32,30);(1,3,16,14,6,18,24,12,7,11,15,2,8,10,5,13)(4,17,31,27,22,32,21,19,23,29,30,9,25,26,20,28);(1,4,6,22,7,23,8,25)(2,9,13,28,14,27,12,19)(3,17,18,32,11,29,10,26)(5,20,16,31,24,21,15,30);(1,5,6,16,7,24,8,15)(2,10,12,11,14,18,13,3)(4,20,22,31,23,21,25,30)(9,26,19,29,27,32,28,17)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,f^b=fi,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72213945, #72231615, … |
| maximal normal subgroup of | #72213937, #72213945, #72231615, … |
| maximal quotient of | #72224893, #72231553, C2×#495573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402769);