Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,16,11,7,13,20,12,15,10,17,27,19,28,21)(8,22,24,31,25,32,35,34,23,14,26,30,36,18,29,33);(2,4)(5,7,10,21,11,15,28,16,12,19,9,20,27,6,13,17)(8,18,24,33,25,22,35,31,23,32,26,34,36,14,29,30);(5,8)(6,14)(7,18)(9,26)(10,24)(11,25)(12,23)(13,29)(15,22)(16,31)(17,30)(19,32)(20,34)(21,33)(27,36)(28,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,e^b=ei,d^c=di,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72237207, #72237224, #72237254, … |
| maximal normal subgroup of | #72237207, #72237224, #72237254, … |
| maximal quotient of | #72237218, #72239327 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53604);