#331155

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

710 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)(9,10,14,17,13,18,12,16)(11,19,22,24,20,15,21,23)(25,26,31,34,30,35,29,36)(27,37,39,33,28,32,38,40);(2,5)(6,8)(9,11)(10,15)(12,21)(13,20)(14,22)(16,24)(17,23)(18,19)(25,27,31,39,30,28,29,38)(26,32,34,40,35,37,36,33);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,14)(15,23)(17,18)(19,24)(25,28,30,27)(26,33,35,40)(29,38,31,39)(32,34,37,36);(3,7)(6,8)(9,12,13,14)(10,17,18,16)(11,21,20,22)(15,23,19,24)(25,29,30,31)(26,34,35,36)(27,38,28,39)(32,40,37,33)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=h,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, 238351);