#411216

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318412);