Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C8 | C23 |
| commutator | C23 | C2×C4×C8 |
| socle | C24 | C22×C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,8)(13,14,17,20,18,21,25,27,16,15,19,22,24,23,26,28);(2,4)(9,10,11,12)(14,15)(20,22)(21,23)(27,28);(5,6,7,8)(13,15,17,22,18,23,25,28,16,14,19,20,24,21,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci176 conjugacy classes — view the full character table.
| maximal subgroup of | #72322577, #72322609, #72322616, … |
| maximal normal subgroup of | #72322577, #72322609, #72322616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54546);