#206701

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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

579 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)(10,14)(11,17)(13,20)(15,22)(18,23)(19,24)(25,26,30,34,29,32,39,33)(27,35,28,37,36,31,38,40);(2,5)(6,8)(9,10)(11,18)(12,14)(13,16)(15,19)(17,22)(20,21)(23,24)(25,27,30,28,29,36,39,38)(26,31,34,40,32,35,33,37);(9,11,13,17,12,19,20,24)(10,15,16,23,14,18,21,22)(25,28,30,36,29,38,39,27)(26,31,33,37,32,35,34,40);(1,3,4,7)(2,6,5,8)(10,14)(11,19)(16,21)(17,24)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=dhi,d^c=dhi

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, 113897);