Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498124 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #22858 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33539 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1765 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,9,12,26)(3,13)(5,14)(6,7)(10,25,28,31)(11,30,24,20)(15,18)(16,23)(17,29,32,27)(21,22);(1,2,7,24,8,12,6,11)(3,10,13,27,16,28,23,29)(4,17,18,25,19,32,15,31)(5,20,14,9,22,30,21,26);(1,3)(2,10)(4,14)(5,15)(6,23)(7,13)(8,16)(9,17)(11,29)(12,28)(18,22)(19,21)(20,31)(24,27)(25,30)(26,32);(1,4)(2,10)(3,14)(5,13)(6,18)(7,15)(8,19)(9,25)(11,27)(12,28)(16,21)(17,30)(20,32)(22,23)(24,29)(26,31)Polycyclic presentation:
2^9|a^2=g,b^2=gh,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=eh,g^b=gi,h^b=hi,f^c=fi,e^d=ehi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72285481, #72285521, … |
| maximal normal subgroup of | #72214039, #72285481, #72285521, … |
| maximal quotient of | #72216448, #72227159, #72227163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405320);