Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,8,12,9,14,19,26,10,13,20,25,21,27,32,35)(7,15,17,28,18,29,31,36,16,11,22,23,30,24,33,34);(5,7)(6,11,13,15)(8,17)(9,18)(10,16)(12,23,25,28)(14,24,27,29)(19,31)(20,22)(21,30)(26,34,35,36)(32,33);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72240009, #72240010, #72240692, … |
| maximal normal subgroup of | #72240009, #72240010, #72240692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53568);