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 | |
1903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,13)(12,22)(14,19)(15,18)(16,20)(17,21)(23,24);(2,6)(5,8)(9,11,16,12)(10,17,20,24)(13,19,22,18)(14,21,15,23);(1,2)(3,8)(4,6)(5,7)(9,12,16,11)(10,13,20,22)(14,21,15,23)(17,19,24,18);(9,13)(10,11)(12,20)(14,24)(15,17)(16,22)(18,21)(19,23);(9,14,16,15)(10,18,20,19)(11,21,12,23)(13,24,22,17)Polycyclic presentation:
2^9|b^2=h,c^2=h,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 | #72339754, #72339788, #72339805, … |
| maximal normal subgroup of | #72339754, #72339788, #72339805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6267966);