#191248

order	512
nilpotency class	3
isoclinism	isoclinic class of #191246
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

578 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)(11,21,25,16)(12,26,28,34)(13,18)(14,20)(15,19)(17,27,33,29)(22,36)(23,32,24,31)(30,37)(35,39)(38,40);(1,2,5,7,6,8,4,3)(9,11,14,24,15,25,13,23)(10,16,20,32,19,21,18,31)(12,22,27,37,28,38,29,39)(17,30,26,40,33,35,34,36);(1,3,4,8,6,7,5,2)(9,12)(10,17,19,33)(11,22)(13,29)(14,27)(15,28)(16,30,21,35)(18,34,20,26)(23,39)(24,37)(25,38)(31,36,32,40);(1,4,6,5)(2,7,8,3)(9,13,15,14)(10,18,19,20)(11,23,25,24)(12,27,28,29)(16,31,21,32)(17,26,33,34)(22,37,38,39)(30,40,35,36)

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=di,g^b=ghi,d^c=dh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98444);