Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| 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 | C22×D42 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,20,16,19);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,17);(7,11)(9,12)(14,19)(15,16)(18,20);(7,11)(9,12);(6,10)(9,12)(14,18)(19,20)Polycyclic presentation:
2^9|a^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231780, #72237566, #72260865, … |
| maximal normal subgroup of | #72231780, #72237566, #72260865, … |
| maximal quotient of | #72235721, #72264170, #72310233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377289);