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×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C23 | C22×C42 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
5214 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)(6,7)(14,15);(5,6,8,7)(13,14,16,15)(17,18,19,20);(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=h,b^2=eg,c^2=i,d^2=efhi,b^a=be,c^a=cf,d^a=de,c^b=cg152 conjugacy classes — view the full character table.
| maximal subgroup of | #72299687, #72299690 |
| maximal normal subgroup of | #72299687, #72299690 |
| maximal quotient of | #72299902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61099);