Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,7,8)(14,18)(15,16)(19,20);(9,10,12,11)(13,14,16,20,17,19,15,18);(2,4);(9,11,12,10)(14,19)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239661, #72239662, #72239671, … |
| maximal normal subgroup of | #72239661, #72239662, #72239671, … |
| maximal quotient of | #72239831, #72286564, #72307247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262113);