Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10,12,16,14,17,13,22)(11,18,20,24,19,15,21,23);(2,6)(3,4)(7,8)(9,11,13,21,14,19,12,20)(10,15,16,23,17,18,22,24);(2,7)(3,4)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=eh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72342988, #72342990, #72343003, … |
| maximal normal subgroup of | #72342988, #72342990, #72343003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14184);