#199193

order	512
nilpotency class	3
isoclinism	isoclinic class of #199182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106389);