Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,24)(22,23);(1,2)(3,6)(4,8)(5,7)(12,21)(16,20)(17,24)(19,22);(9,11,13,18)(10,14,15,23)(12,22,21,19)(16,24,20,17);(10,15)(14,23)(16,20)(17,24);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^9|c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257606, #72257658, #72283314 |
| maximal normal subgroup of | #72257606, #72257658, #72283314 |
| maximal quotient of | #72264035 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256182);