Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
616 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)(5,6,8,11,9,12,7,10)(13,14)(15,25)(16,20)(17,22)(18,21)(19,24)(23,27)(26,28);(2,4)(6,10,12,11)(13,15,18,20,16,21,25,14,19,22,28,23,27,26,17,24);(13,16,19,27)(14,20,24,23)(15,26,22,21)(17,18,25,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210543 |
| maximal normal subgroup of | #72210543 |
| maximal quotient of | #72210445 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33967);