Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13894 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,9,5,11)(3,8,13,23,15,22,16,20)(4,10,18,25,17,24,19,26)(14,28,29,32,30,21,27,31);(1,3)(2,8,9,22)(4,14)(5,13)(6,16)(7,15)(10,21,24,28)(11,23,12,20)(17,30)(18,27)(19,29)(25,32,26,31);(2,9)(3,13)(5,6)(8,20)(10,24)(14,27)(15,16)(18,19)(21,31)(22,23)(28,32)(29,30);(1,4)(2,10,9,24)(3,14)(5,18)(6,19)(7,17)(8,21,22,28)(11,25,12,26)(13,27)(15,30)(16,29)(20,31,23,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268514);