Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383753 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12)(3,8)(4,15,18,28)(5,21)(7,24,23,11)(9,20,29,19)(10,22)(13,27)(14,30)(16,32)(17,26)(25,31);(1,3)(2,8)(4,16)(5,6)(7,22)(9,26)(10,24)(11,30)(12,21)(13,20)(14,23)(15,32)(17,19)(18,31)(25,28)(27,29);(1,4,6,18)(2,9)(3,13,5,17)(7,19,23,20)(8,25)(10,27)(11,15)(12,29)(14,31,22,16)(21,32)(24,28)(26,30);(1,5,6,3)(2,10)(4,17,18,13)(7,22,23,14)(8,24)(9,27)(11,21)(12,30)(15,32)(16,20,31,19)(25,28)(26,29)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216047, #72216071, … |
| maximal normal subgroup of | #72213409, #72216047, #72216071, … |
| maximal quotient of | C2×#383753, #72250488, #72252187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290949);