Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384259 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2333 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14452 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,21)(3,12,16,28,14,8,15,24)(4,17,19,26,20,9,22,27)(13,29,18,25,30,23,31,32);(1,3,5,16,7,14,6,15)(2,8,21,24,10,12,11,28)(4,18,19,13,20,31,22,30)(9,25,27,29,17,32,26,23);(1,4,5,19,7,20,6,22)(2,9,21,27,10,17,11,26)(3,13,15,18,14,30,16,31)(8,23,28,32,12,29,24,25);(8,12)(23,29)(24,28)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=eg,g^2=i,h^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,f^b=fi,g^b=gi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251696, #72251701, #72251711, … |
| maximal normal subgroup of | #72251696, #72251701, #72251711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294378);