#193331

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
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

869 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)(3,7)(6,8)(9,10,14,19,15,18,13,26)(11,20,22,16,23,31,27,30)(12,17,24,34,25,32,28,33)(21,36,39,40,38,35,37,29);(9,11,14,22,15,23,13,27)(10,16,19,31,18,30,26,20)(12,21,24,39,25,38,28,37)(17,29,34,36,32,40,33,35);(1,2,4,6)(3,8,7,5)(9,12)(10,17)(11,21)(13,28)(14,24)(15,25)(16,29)(18,32)(19,34)(20,35)(22,39)(23,38)(26,33)(27,37)(30,40)(31,36);(10,18)(16,30)(17,32)(19,26)(20,31)(29,40)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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,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, 100527);