#415555

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322751);