Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,7,10,8,9,11,12);(2,4)(13,14,15,18,16,17,19,20);(14,17)(18,20);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=be,d^a=dh,c^b=ci152 conjugacy classes — view the full character table.
| maximal subgroup of | #72224709, #72224710, #72224711, … |
| maximal normal subgroup of | #72224709, #72224710, #72224711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88051);