#250345

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157541);