Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #386 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,9,8,10,11,12)(13,14)(15,22,24,18)(16,19)(17,20)(21,26)(23,28,25,27);(1,2)(3,4)(13,15,17,24)(14,18,20,22)(16,25,26,23)(19,28,21,27);(2,4)(14,19,20,21)(18,27,22,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,g^b=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72267644 |
| maximal normal subgroup of | #72267644 |
| maximal quotient of | #72267649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25869);