Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×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 | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4126 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)(13,14,17,19)(15,18,16,20);(2,4)(13,15,17,16)(14,18,19,20);(9,11,12,10)(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264369, #72264377, #72264411, … |
| maximal normal subgroup of | #72264369, #72264377, #72264411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262006);