Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10,13,17,14,16,12,22)(11,18,21,23,19,24,20,15);(1,3,5,8,6,2,4,7)(9,11)(10,15,16,23)(12,20)(13,21)(14,19)(17,18,22,24);(10,16)(12,13)(15,23)(20,21);(1,4,6,5)(2,7,3,8)(10,16)(11,19)(17,22)(20,21)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=di,f^b=fh,e^c=eh,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239301, #72257603, #72257606, … |
| maximal normal subgroup of | #72239301, #72257603, #72257606, … |
| maximal quotient of | #72239304 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103484);