Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(9,10)(11,12)(13,14,17,19,16,18,15,20);(1,2,4,3)(5,6,7,8)(9,10)(11,12)(13,14,15,20,16,18,17,19);(6,8)(10,12)(14,18)(19,20);(1,3,4,2)(13,15,16,17)(14,19,18,20)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305678, #72305685, #72305689, … |
| maximal normal subgroup of | #72305678, #72305685, #72305689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274305);