#218609

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(3,7)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,22,24,23)(25,26,29,33,30,28,34,32)(27,31,36,39,37,38,35,40);(1,2,4,5)(3,6,7,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,23,17,22)(25,27,34,35,30,37,29,36)(26,31,32,40,28,38,33,39);(2,5)(6,8)(10,15)(11,19)(13,21)(14,22)(16,17)(18,23)(25,28)(26,30)(27,31)(29,32)(33,34)(35,40)(36,39)(37,38);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,17)(20,24)(25,29,30,34)(26,32,28,33)(27,35,37,36)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125805);