Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
752 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)(3,7)(4,6)(5,8)(9,10,12,16,13,14,19,21)(11,15,17,22,18,20,23,24);(2,6,8,7)(10,14)(15,20);(1,3,5,4)(2,7,8,6)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,e^b=eh,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266619 |
| maximal normal subgroup of | #72266619 |
| maximal quotient of | #72266600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34404);