Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461207 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16383 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,12,7,13)(3,14,17,9,18,28,16,26)(4,19,5,22,21,10,23,27)(15,25,20,24,32,29,31,30);(1,3)(2,9,12,26)(4,20)(5,15)(6,16)(7,17)(8,18)(10,25,19,29)(11,14,13,28)(21,31)(22,24,27,30)(23,32);(1,4)(2,10)(3,15)(5,7)(6,23)(8,21)(9,24)(11,22)(12,19)(13,27)(14,29)(16,20)(17,31)(18,32)(25,28)(26,30);(1,5,7,4,8,23,6,21)(2,10,13,22,12,19,11,27)(3,15,17,20,18,32,16,31)(9,25,28,24,26,29,14,30)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368404);