Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,11,14,12)(13,18,20,19);(5,6,8,12,10,14,9,11)(7,13,15,19,16,20,17,18);(6,12)(7,15)(8,9)(11,14)(16,17)(18,19);(1,3,4,2)(5,7,8,15,10,16,9,17)(6,13,12,19,14,20,11,18)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224750, #72224755, #72224757 |
| maximal normal subgroup of | #72224750, #72224755, #72224757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288892);