Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495197 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1214 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)(3,5,13,19)(4,14)(6,20)(7,10)(8,17)(9,21)(11,23,24,32)(12,26,27,18)(15,28)(16,31,29,25)(22,30);(1,3,5,10,7,13,19,2)(4,11,16,22,15,24,29,8)(6,12,18,20,21,27,26,9)(14,17,25,23,28,30,31,32);(1,4,6,17,7,15,21,30)(2,8,9,14,10,22,20,28)(3,11,12,25,13,24,27,31)(5,16,18,23,19,29,26,32);(4,15)(8,22)(11,24)(14,28)(16,29)(17,30)(23,32)(25,31)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72236890, #72236900, #72237542, … |
| maximal normal subgroup of | #72236890, #72236900, #72237542, … |
| maximal quotient of | #72311871, #72312860, C2×#495198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402394);