Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,13,9,12,18,11)(7,14,15,19,16,10,17,20);(1,2)(3,4)(5,7)(6,10)(8,17)(9,16)(11,20)(12,14)(13,19)(15,18);(2,4)(6,11)(8,18)(10,19)(12,13)(14,20)(15,17);(5,7)(6,10,12,14)(8,15)(9,16)(11,19,13,20)(17,18)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72219643, #72219665, #72219671, … |
| maximal normal subgroup of | #72219643, #72219665, #72219671, … |
| maximal quotient of | #72260577 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327027);