Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,11,15,21,32,12,17,20,13,9,27,7,18)(8,14,24,29,25,30,22,35,26,31,34,28,23,36,19,33);(5,7,9,20,12,21,11,10)(6,13,15,18,17,16,27,32)(8,22,23,24,26,19,25,34)(14,29,30,35,31,28,36,33);(1,2,3,4);(5,8,10,24,11,25,21,22,12,26,20,34,9,23,7,19)(6,14,16,29,15,30,32,35,17,31,13,28,27,36,18,33)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225220, #72225223, #72225458, … |
| maximal normal subgroup of | #72225220, #72225223, #72225458, … |
| maximal quotient of | #72225220 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403288);