#334815

order	512
nilpotency class	3
isoclinism	isoclinic class of #334695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242011);