#332643

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24180	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239839);