Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24208 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24);(2,6)(7,8)(10,14)(11,12)(18,21)(19,24)(22,23);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(18,21)(19,24)(22,23);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(18,22)(21,23);(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^9|c^2=fgi,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72305437, #72305450, #72305468 |
| maximal normal subgroup of | #72305437, #72305450, #72305468 |
| maximal quotient of | #72305631, #72305772 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6532167);