Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10,15,19)(11,20,22,12)(13,18,14,17)(16,23,24,21);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,18)(13,21)(14,23)(15,22)(17,20)(19,24);(2,7)(3,4)(6,8)(9,11)(10,12)(13,23)(14,21)(15,22)(16,18)(17,24)(19,20);(9,12)(10,11)(13,16)(14,24)(15,20)(17,21)(18,23)(19,22);(9,13,15,14)(10,17,19,18)(11,21,22,23)(12,16,20,24)Polycyclic presentation:
2^9|a^2=hi,b^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338311, #72338436, #72338440 |
| maximal normal subgroup of | #72338311, #72338436, #72338440 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267985);