#119663

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

667 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26859);