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) = C23×C4 |
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 | C24 | C22×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(11,21)(22,31)(23,32)(33,36);(5,6,8,13,9,14,18,26,10,12,19,24,20,25,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(1,3,4,2)(5,7)(6,11)(8,16)(9,17)(10,15)(12,21)(13,22)(14,23)(18,29)(19,27)(20,28)(24,31)(25,32)(26,33)(30,35)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,g^2=h,h^2=i,b^a=bd,c^a=ce,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72319978, #72320017, #72320029, … |
| maximal normal subgroup of | #72319978, #72320017, #72320029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54313);