Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459714 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16347 |
| 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 | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,9,6,11)(3,13,16,8,17,26,14,23)(4,18,19,24,21,25,20,10)(15,22,29,31,30,32,28,27);(1,3)(2,8,9,23)(4,15)(5,16)(6,14)(7,17)(10,22,24,32)(11,13,12,26)(18,31,25,27)(19,29)(20,28)(21,30);(2,9)(3,14)(4,19)(5,6)(8,13)(10,18)(16,17)(20,21)(22,32)(23,26)(24,25)(28,29);(1,4,5,19,7,21,6,20)(2,10,12,18,9,24,11,25)(3,15,16,29,17,30,14,28)(8,22,26,31,23,32,13,27)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366910);