#411218

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=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, 318414);