Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,10)(8,11)(9,12)(13,14,15,18,16,17,19,20);(1,2,3,4)(6,10)(7,8)(11,12);(6,11)(7,8)(10,12)(14,17)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=dh,e^b=eh,d^c=dh,e^c=eh,f^c=fhi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257922, #72257924, #72257931, … |
| maximal normal subgroup of | #72257922, #72257924, #72257931, … |
| maximal quotient of | #72264879, #72308704 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13365);