Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500938 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24338 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
705 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)(3,12,15,31)(4,18)(5,22)(6,26)(7,11)(8,16,28,13)(9,19)(10,21)(14,24,29,20)(17,30,27,32)(23,25);(3,13,15,16)(4,19)(5,23)(6,21)(8,12,28,31)(9,18)(10,26)(14,32,29,30)(17,24,27,20)(22,25);(1,3)(2,8)(4,20)(5,24)(6,27)(7,15)(9,30)(10,14)(11,28)(12,22)(13,19)(16,23)(17,21)(18,31)(25,32)(26,29);(1,2)(3,14,15,29)(4,9)(5,25)(6,10)(7,11)(8,27,28,17)(12,20,31,24)(13,32,16,30)(18,19)(21,26)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=e,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fhi,g^c=ghi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72307725, C2×#500938 |
| maximal normal subgroup of | #72213945, #72307725, C2×#500938 |
| maximal quotient of | #72226807, #72239902, #72250668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408134);