#411395

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

757 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,26)(13,20)(14,18)(15,19)(16,23)(17,27)(22,36)(24,32)(25,31)(28,35)(29,34)(30,38)(33,39)(37,40);(2,5)(6,8)(9,11,14,25,15,24,13,23)(10,16,20,32,19,31,18,21)(12,22,29,39,27,37,28,38)(17,33,34,36,26,30,35,40);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,27,14,29)(10,17,18,35,19,26,20,34)(11,22,23,38,24,37,25,39)(16,30,21,36,31,33,32,40);(10,18,19,20)(12,27)(16,21,31,32)(17,34,26,35)(22,37)(28,29)(30,40,33,36)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318591);