Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,13,10,14,8,11)(7,15,16,12,18,19,17,20);(1,2)(3,4)(6,11,14,13)(12,19,20,15);(2,4)(5,6)(7,12)(8,13)(9,11)(10,14)(15,16)(17,19)(18,20);(5,7,8,16,10,18,9,17)(6,12,13,15,14,20,11,19)Polycyclic presentation:
2^9|a^2=f,b^2=eh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224752, #72224756, #72224757 |
| maximal normal subgroup of | #72224752, #72224756, #72224757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313304);