Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,27,22,24,29,32,23,28,19,25,20,26,30,31);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(18,24,28,26)(19,21)(25,32,27,31)(29,30);(17,19,22,30,23,21,20,29)(18,25,24,31,28,27,26,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,f^a=fi,c^b=ci,e^b=egi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285281, #72285353, #72285367, … |
| maximal normal subgroup of | #72285281, #72285353, #72285367, … |
| maximal quotient of | #72286160 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34237);