#199189

order	512
nilpotency class	3
isoclinism	isoclinic class of #199182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#60559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1128 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,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, 106385);