Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16893 |
| 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 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,9)(4,18,21,28)(5,16)(6,15,8,13)(10,23,27,20)(11,24)(14,30)(17,22)(19,29)(25,26)(31,32);(1,3,6,17,7,16,8,11)(2,9,13,24,12,5,15,22)(4,14,23,29,21,31,20,26)(10,25,28,32,27,19,18,30);(1,4,6,20,7,21,8,23)(2,10,13,18,12,27,15,28)(3,14,17,26,16,31,11,29)(5,19,22,32,9,25,24,30);(1,5,7,9)(2,11,12,17)(3,15,16,13)(4,19,21,25)(6,22,8,24)(10,26,27,29)(14,18,31,28)(20,32,23,30)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381219);