Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667700 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,9)(7,11,12,10)(13,14,17,20)(15,18,16,19);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,16)(14,18,20,19);(2,4)(6,9)(10,11)(14,19)(15,16)(18,20);(5,7,8,12)(6,10,9,11);(2,4)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280286, #72280484, #72280773, … |
| maximal normal subgroup of | #72280286, #72280484, #72280773, … |
| maximal quotient of | #72304945, #72305564, #72305760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578142);