#193010

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

863 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100206);