Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,11,9,7,10,12)(14,19,21,25)(15,22)(20,27,28,26)(23,24);(1,2,3,4)(13,14)(15,20)(16,19)(17,25)(18,21)(22,28)(23,26)(24,27);(5,7,8,12,9,6,10,11)(13,15,17,24,18,22,16,23)(14,20,25,27,21,28,19,26);(13,16,18,17)(14,19,21,25)(15,23,22,24)(20,26,28,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,e^b=eh,f^b=fh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72285948, #72285998, #72286341, … |
| maximal normal subgroup of | #72285948, #72285998, #72286341, … |
| maximal quotient of | #72286564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90626);