#250280

order	512
nilpotency class	3
isoclinism	isoclinic class of #250199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24231	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157476);