Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497868 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #22670 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,9,13,10)(3,5)(6,7)(11,25,29,27)(12,21,23,26)(14,22)(15,16)(17,30,32,28)(18,31)(19,24);(1,2)(3,10)(4,17)(5,21)(6,12)(7,23)(8,13)(9,16)(11,18)(14,29)(15,26)(19,25)(20,32)(22,30)(24,27)(28,31);(1,3,6,5,8,16,7,15)(2,10,12,21,13,9,23,26)(4,14,19,22,20,18,24,31)(11,27,28,17,29,25,30,32);(1,4)(2,11)(3,14)(5,22)(6,19)(7,24)(8,20)(9,25)(10,27)(12,28)(13,29)(15,31)(16,18)(17,21)(23,30)(26,32)Polycyclic presentation:
2^9|a^2=g,c^2=f,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214153, #72214178, … |
| maximal normal subgroup of | #72214039, #72214153, #72214178, … |
| maximal quotient of | #72216388, C3×#497868, #72226987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405064);