Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388376 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14519 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,24,18)(3,12,16,29,15,8,22,27)(4,5,9,21,19,20,23,30)(13,25,28,32,17,14,26,31);(1,3)(2,8)(4,17)(5,14)(6,22)(7,15)(9,28)(10,12)(11,29)(13,19)(16,24)(18,27)(20,25)(21,32)(23,26)(30,31);(1,4,6,23,7,19,24,9)(2,5,18,21,10,20,11,30)(3,13,16,26,15,17,22,28)(8,25,29,32,12,14,27,31);(1,5)(2,9)(3,14)(4,18)(6,21)(7,20)(8,26)(10,23)(11,19)(12,28)(13,27)(15,25)(16,31)(17,29)(22,32)(24,30)Polycyclic presentation:
2^9|a^2=h,c^2=eh,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72320059, #72320069, #72320075, … |
| maximal normal subgroup of | #72320059, #72320069, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295573);