Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,17,20,14)(12,16)(13,15)(18,23,19,22)(21,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,20)(15,17)(16,22)(19,21)(23,24);(3,8)(5,7)(11,18)(12,21)(14,22)(16,24)(17,23)(19,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,f^c=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214356, #72214640, #72214821, … |
| maximal normal subgroup of | #72214356, #72214640, #72214821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20292);