#176280

order	512
nilpotency class	3
isoclinism	isoclinic class of #176173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

676 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83476);