Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469932 |
| 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 |
| prime radical | #36644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,24,21,19,20,23);(11,12)(14,15)(17,19,21,18)(20,23,22,24);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15);(9,12,13,11)(10,15,16,14);(1,4,5,3)(2,6,7,8)(10,16)(14,15)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281158, #72281391, #72281554, … |
| maximal normal subgroup of | #72281158, #72281391, #72281554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396306);