Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384163 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14373 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1231 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)(5,18,19,9)(6,10)(12,20,15,25)(13,24)(16,29,30,22)(17,23)(21,27)(26,31)(28,32);(1,3)(2,7)(4,15)(5,13)(6,14)(8,20)(9,24)(10,11)(12,17)(16,28)(18,21)(19,27)(22,32)(23,25)(26,30)(29,31);(1,4)(2,8)(3,12,14,15)(5,16)(6,17)(7,20,11,25)(9,22)(10,23)(13,26,27,28)(18,29)(19,30)(21,31,24,32);(1,5)(2,9)(3,13)(4,16)(6,19)(7,21)(8,22)(10,18)(11,24)(12,26)(14,27)(15,28)(17,30)(20,31)(23,29)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237749, #72237754, #72237759, … |
| maximal normal subgroup of | #72237749, #72237754, #72237759, … |
| maximal quotient of | #72265871 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291359);