#250529

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1325 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157725);