#118223

order	512
nilpotency class	3
isoclinism	isoclinic class of #118219
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

584 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,e^b=ehi,f^b=fh,d^c=di,e^c=eh,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25419);