#314826

order	512
nilpotency class	3
isoclinism	isoclinic class of #262787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

577 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ei,c^2=i,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,d^c=dh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222022);