#208482

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

861 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115678);