Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
3790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(13,14)(15,17,19,20)(16,18);(2,4)(13,15)(14,17,18,20)(16,19);(6,8)(9,10,11,12)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce,d^a=di,g^a=gi,d^b=di,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72234290, #72260737, #72260780, … |
| maximal normal subgroup of | #72234290, #72260737, #72260780, … |
| maximal quotient of | #72234300, #72267709, #72280379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30648);