Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13956 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2599 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)(3,11,14,24)(4,15,17,29)(5,9)(6,10)(7,13,19,28)(8,16,21,30)(12,25)(18,22)(20,27)(23,31)(26,32);(1,3)(2,7,10,19)(4,12)(5,13)(6,14)(8,20,21,25)(9,11,22,24)(15,23,29,32)(16,26)(17,27)(18,28)(30,31);(5,18)(7,19)(9,22)(13,28)(15,29)(16,30)(20,25)(26,31);(1,4)(2,8,10,21)(3,12)(5,16)(6,17)(7,20,19,25)(9,15,22,29)(11,23,24,32)(13,26)(14,27)(18,30)(28,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72251414, #72251422, #72251427, … |
| maximal normal subgroup of | #72251414, #72251422, #72251427, … |
| maximal quotient of | #72255866 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271101);