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 | #14513 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1019 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)(3,13,15,8)(4,9,17,23)(5,18)(6,22)(7,11)(10,20)(12,25)(14,26,19,30)(16,31,32,27)(21,28)(24,29);(1,3)(2,8)(4,12)(5,19)(6,23)(7,15)(9,25)(10,26)(11,13)(14,20)(16,29)(17,22)(18,30)(21,31)(24,27)(28,32);(1,4,6,8,7,17,25,13)(2,9,12,15,11,23,22,3)(5,16,24,26,20,32,21,30)(10,27,29,19,18,31,28,14);(1,5,6,24,7,20,25,21)(2,10,12,29,11,18,22,28)(3,14,9,27,15,19,23,31)(4,16,8,26,17,32,13,30)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=efg,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295282);