#193342

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

861 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fh,e^c=eh,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, 100538);