Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510874 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32002 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #28180 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,18,25)(3,11,15,28,13,7,21,24)(4,16,19,32,20,9,17,27)(12,26,31,29,30,22,14,23);(1,3)(2,7)(4,12)(5,21)(6,13)(8,11)(9,26)(10,28)(14,19)(15,18)(16,22)(17,31)(20,30)(23,27)(24,25)(29,32);(2,8)(3,12,13,30)(4,17)(5,18)(7,22,11,26)(9,27)(14,21,31,15)(16,32)(19,20)(23,24,29,28);(7,11)(22,26)(23,29)(24,28)Polycyclic presentation:
2^9|a^2=h,c^2=ef,g^2=hi,h^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=gh,h^b=hi,g^c=ghi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#510874, #72320520, … |
| maximal normal subgroup of | #72237280, C2×#510874, #72320520, … |
| maximal quotient of | C2×#510874, #72250941, #72257135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418070);