#332888

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
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	#10176	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240084);