Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35680 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17)(11,18,20,24)(12,14,21,16)(15,22,23,19);(1,3)(2,5)(4,7)(6,8)(10,14)(12,21)(15,18)(16,17)(19,22)(23,24);(5,8)(12,21)(14,16)(18,24)(19,22);(9,11)(10,15)(12,22)(13,20)(14,18)(16,24)(17,23)(19,21)Polycyclic presentation:
2^9|a^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,e^c=eh,e^d=eh,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72231737 |
| maximal normal subgroup of | #72231737 |
| maximal quotient of | #72231648 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155844);