Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14373 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,7,12,26,11)(3,14,19,29,18,30,17,8)(4,9,21,24,10,5,22,25)(15,31,28,27,20,16,32,23);(1,3)(2,8)(4,20)(5,23)(6,19)(7,18)(9,27)(10,15)(11,30)(12,29)(13,14)(16,24)(17,26)(21,32)(22,28)(25,31);(1,4)(2,9)(3,15,18,20)(5,12)(6,21)(7,10)(8,27,29,23)(11,25)(13,24)(14,16,30,31)(17,32,19,28)(22,26);(1,5)(2,10)(3,16)(4,12)(6,25)(7,9)(8,28)(11,21)(13,22)(14,20)(15,30)(17,27)(18,31)(19,23)(24,26)(29,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291366);