Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,10,15,11,16,9,12)(7,8,14,13,18,20,17,19);(1,2)(3,4)(6,12,16,15)(8,19,20,13);(5,7,10,14,11,18,9,17)(6,13,15,20,16,19,12,8);(2,4)(5,8,10,13,11,20,9,19)(6,14,15,18,16,17,12,7)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=dei,f^c=fi,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72215961, #72219684, #72219685, … |
| maximal normal subgroup of | #72215961, #72219684, #72219685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263877);