Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,13)(7,11,14,19)(8,15,12,9)(16,17,20,18);(6,8)(7,14)(9,15)(11,16)(12,13)(19,20);(5,7,15,17,10,14,9,18)(6,11,8,16,13,19,12,20);(5,8,15,13,10,12,9,6)(7,16,17,19,14,20,18,11)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ef,e^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,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238735 |
| maximal normal subgroup of | #72238735 |
| maximal quotient of | #72238748 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292722);