#250409

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

1098 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)(3,4)(7,8)(9,10,15,20)(11,21,25,31)(12,26,29,33)(13,19,14,17)(16,24,32,22)(18,28,34,27)(23,30,38,40)(35,36,39,37);(1,2)(3,8)(4,6)(5,7)(9,11,15,25)(10,16,20,32)(12,23,29,38)(13,24,14,22)(17,31,19,21)(18,30,34,40)(26,35,33,39)(27,36,28,37);(1,2)(3,6)(4,8)(5,7)(10,17)(11,22)(13,14)(18,33)(19,20)(21,31)(23,36)(24,25)(26,34)(27,28)(35,39)(37,38);(1,2)(3,8)(4,6)(5,7)(9,12,15,29)(10,18,20,34)(11,23,25,38)(13,28,14,27)(16,30,32,40)(17,33,19,26)(21,35,31,39)(22,36,24,37)

Polycyclic presentation:

2^9|a^2=i,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, 157605);