Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | D4×#121 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2692 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)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14,17,15)(16,18,20,19);(2,4)(14,18)(15,19)(16,20);(5,7,8,11)(6,10,9,12)(13,15,17,14)(16,19,20,18);(14,18)(15,19)(16,20)Polycyclic presentation:
2^9|b^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72239637, #72256943, #72257026, … |
| maximal normal subgroup of | #72239637, #72256943, #72257026, … |
| maximal quotient of | #72239684, #72249749, #72264559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482452);