#191236

order	512
nilpotency class	3
isoclinism	isoclinic class of #191234
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 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)(3,5)(4,7)(6,8)(9,10)(11,21,24,16)(12,25,27,34)(13,18)(14,20)(15,19)(17,26,32,29)(22,36)(23,33,28,31)(30,38)(35,37)(39,40);(1,2,5,7,6,8,4,3)(9,11,14,23,15,24,13,28)(10,16,20,31,19,21,18,33)(12,22,26,38,27,39,29,37)(17,30,25,40,32,35,34,36);(1,3,4,8,6,7,5,2)(9,12)(10,17,19,32)(11,22)(13,29)(14,26)(15,27)(16,30,21,35)(18,34,20,25)(23,38)(24,39)(28,37)(31,40,33,36);(1,4,6,5)(2,7,8,3)(9,13,15,14)(10,18,19,20)(11,23,24,28)(12,26,27,29)(16,31,21,33)(17,25,32,34)(22,37,39,38)(30,36,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98432);