Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| 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×D4×#41 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4317 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)(13,14,16,18,17,20,15,19);(2,4)(5,6,8,9)(7,10,11,12)(14,18)(15,16)(19,20);(6,9)(7,11)(14,19)(15,16)(18,20);(5,7)(6,10)(8,11)(9,12);(6,9)(7,11)Polycyclic presentation:
2^9|a^2=fi,b^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
| maximal subgroup of | #72249682, #72251233, #72251252, … |
| maximal normal subgroup of | #72249682, #72251233, #72251252, … |
| maximal quotient of | #72249749, #72249758, #72264261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503436);