#147880

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,g^b=gi,h^b=hi,d^c=dhi,e^c=ehi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55076);