#148040

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,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, 55236);