Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200418 |
| 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 | #35050 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,21,28,4,18)(3,13,17,27,31,32,15,25)(6,12,20,29,7,9,22,10)(8,16,26,23,24,14,30,19);(1,3)(2,8,12,27)(4,19)(5,23)(6,14)(7,16)(9,25,28,24)(10,26,18,32)(11,13,29,30)(15,20)(17,22)(21,31);(2,9)(3,14)(4,20)(5,22)(8,24)(12,28)(13,30)(15,17)(16,31)(19,23)(25,27)(26,32);(2,9)(8,25)(10,11)(12,28)(13,26)(18,29)(24,27)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107860);