#415881

order	512
nilpotency class	3
isoclinism	isoclinic class of #394271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=eh,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi

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, 323077);