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 | #33541 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1765 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,9,14,8,12)(3,15,18,5,4,19,17,23)(6,24,27,21,29,30,28,11)(10,25,26,16,22,32,31,20);(1,3)(2,10,14,22)(4,9)(5,21,23,11)(6,25)(7,17)(8,18)(12,26,13,31)(15,30,19,24)(16,28)(20,27)(29,32);(1,4)(2,5)(3,9)(6,20)(7,17)(8,18)(10,21)(11,22)(12,15)(13,19)(14,23)(16,29)(24,31)(25,27)(26,30)(28,32);(1,5)(2,4)(3,14)(6,26)(7,19)(8,15)(9,23)(10,28)(11,25)(12,18)(13,17)(16,24)(20,30)(21,32)(22,27)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=i,g^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, #72227161, #72227162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405322);