#331177

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh,g^c=gi

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, 238373);