Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533087 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(13,14,17,18,16,20,15,19);(5,6)(7,10)(8,9)(11,12)(14,18)(15,17)(19,20);(2,4)(6,9)(10,12);(5,7)(6,10)(8,11)(9,12);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=fgi,f^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,e^d=eh89 conjugacy classes — view the full character table.
| maximal subgroup of | #72279976 |
| maximal normal subgroup of | #72279976 |
| maximal quotient of | #72306966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449505);