Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,10,14,11,15,9,19)(7,16,8,13,17,20,18,12);(5,7,9,18,11,17,10,8)(6,12,14,16,15,13,19,20);(1,2,3,4);(5,8,10,17,11,18,9,7)(6,13,14,20,15,12,19,16);(7,17)(8,18)(12,13)(16,20)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,e^b=ei,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224852 |
| maximal normal subgroup of | #72224852 |
| maximal quotient of | #72224842 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280960);