Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14641 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,8)(5,10)(6,9)(12,26,19,20)(13,22,16,21)(15,23)(17,24)(18,25)(27,31,28,30)(29,32);(3,12)(4,15)(5,18)(7,20)(10,25)(11,26)(13,28)(14,19)(16,27)(21,30)(22,31)(24,32);(1,3)(2,7)(4,16)(5,19)(6,14)(8,22)(9,11)(10,26)(12,18)(13,15)(17,27)(20,25)(21,23)(24,30)(28,29)(31,32);(1,4,5,17,6,15,18,29)(2,8,10,24,9,23,25,32)(3,13,12,27,14,16,19,28)(7,21,20,30,11,22,26,31)Polycyclic presentation:
2^9|a^2=h,d^2=g,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338910, #72338917, #72338919 |
| maximal normal subgroup of | #72338910, #72338917, #72338919 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298756);