Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #33175 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,14,11)(12,13,16,15)(18,21)(19,23)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11,13,16,14,10,15,12)(18,22)(21,24);(3,7)(5,8)(17,18,19,21,20,22,23,24);(9,12,13,11,14,16,15,10)Polycyclic presentation:
2^9|a^2=ghi,b^2=e,c^2=f,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72224773, #72224777, #72224779 |
| maximal normal subgroup of | #72224773, #72224777, #72224779 |
| maximal quotient of | #72224930, #72224938 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 222950);