Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C22 | C2×C43 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
5438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,7)(13,14,16,15)(18,20);(6,7)(9,10,12,11);(13,14,16,15)(17,18,19,20);(1,3,4,2)(5,7,8,6)(9,11,12,10)(13,15,16,14)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=gh,b^a=be,c^a=cf,d^b=de200 conjugacy classes — view the full character table.
| maximal subgroup of | #72299159, #72299186, #72299194 |
| maximal normal subgroup of | #72299159, #72299186, #72299194 |
| maximal quotient of | #72299915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60923);