Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254467 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #32890 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,17,19,15)(12,16,21,22)(18,20,23,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20,21,24)(13,19)(14,17)(16,23,22,18);(5,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,20)(17,24)(19,23);(9,11)(10,15)(12,18)(13,19)(14,17)(16,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72255865, #72255909, #72255929, … |
| maximal normal subgroup of | #72255865, #72255909, #72255929, … |
| maximal quotient of | #72263247 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161681);