#349637

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
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×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256833);