#250415

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

1078 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)(9,10,15,11)(12,16,24,19)(13,17,26,20)(14,21,29,18)(22,28,31,25)(23,33,36,30)(27,34,38,32)(35,37,40,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,10)(12,22,24,31)(13,20,26,17)(14,18,29,21)(16,25,19,28)(23,35,36,40)(27,32,38,34)(30,37,33,39);(1,2)(3,6)(4,8)(5,7)(9,12,15,24)(10,16,11,19)(13,23,26,36)(14,28,29,25)(17,30,20,33)(18,22,21,31)(27,39,38,37)(32,35,34,40);(1,2)(3,8)(4,6)(5,7)(9,13)(10,17)(11,20)(12,23)(14,27)(15,26)(16,30)(18,32)(19,33)(21,34)(22,35)(24,36)(25,37)(28,39)(29,38)(31,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 157611);