Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17145 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,19,8)(3,12,28,21,13,7,15,23)(4,9,20,26,18,27,17,24)(14,22,16,32,30,29,31,25);(1,3,5,15,6,13,19,28)(2,7,11,21,10,12,8,23)(4,16,17,30,18,31,20,14)(9,25,24,22,27,32,26,29);(2,8)(3,13)(5,19)(7,21)(9,26)(10,11)(12,23)(14,30)(17,20)(22,32)(24,27)(25,29);(1,4)(2,9)(3,14)(5,20)(6,18)(7,22)(8,24)(10,27)(11,26)(12,29)(13,30)(15,31)(16,28)(17,19)(21,25)(23,32)Polycyclic presentation:
2^9|a^2=fg,b^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251786, #72251792, #72251794, … |
| maximal normal subgroup of | #72251786, #72251792, #72251794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385583);