#293207

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,18,20,15)(12,22,17,14)(19,24,23,21)(28,36)(30,38)(32,37)(35,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,23)(13,20)(14,21)(16,18)(17,19)(22,24)(25,26,29,31)(27,33,34,40)(28,37,36,32)(30,39,38,35);(9,12,14,10,13,17,22,16)(11,19,21,18,20,23,24,15)(25,27,29,34)(26,30)(28,35,36,39)(31,38)(32,33)(37,40);(2,6)(5,7)(10,16)(11,20)(12,17)(21,24)(27,34)(30,38)(33,40)(35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

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, 200403);