Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15)(17,19,20,28,21,27,29,26)(18,22,24,32,23,30,31,25);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16);(2,8)(4,12)(6,14)(9,15)(18,23)(22,30)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104535);