Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16851 |
| 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 | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,9)(3,13,16,22,17,23,15,8)(4,10,19,24,20,25,18,26)(14,28,30,32,31,27,29,21);(1,3,6,16,7,17,5,15)(2,8,9,23,12,22,11,13)(4,14,18,29,20,31,19,30)(10,21,24,28,25,32,26,27);(2,9)(4,18)(5,6)(8,13)(10,25)(11,12)(14,29)(15,16)(19,20)(22,23)(27,28)(30,31);(1,4)(2,10,12,25)(3,14)(5,19)(6,18)(7,20)(8,21,22,32)(9,24,11,26)(13,27,23,28)(15,30)(16,29)(17,31)Polycyclic presentation:
2^9|a^2=f,b^2=egh,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380354);