Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(9,10,12,11)(13,14,16,18,17,19,15,20);(2,4)(6,8)(9,11,12,10);(5,6,7,8)(14,18)(15,16)(19,20);(2,4)(14,19)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72305569, #72305574, #72305593, … |
| maximal normal subgroup of | #72305569, #72305574, #72305593, … |
| maximal quotient of | #72309878 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275649);