Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1781 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)(10,18)(13,19);(1,3,4,2)(5,6)(7,13)(8,16)(9,11)(10,15)(12,17)(14,19)(18,20);(5,7,8,15,9,14,17,20)(6,10,12,13,11,18,16,19);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,17)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72221417, C4×#23882, #72252452, … |
| maximal normal subgroup of | C4×#23882, #72252452, #72252528, … |
| maximal quotient of | #72221417, #72257933, #72263993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275279);