#250798

order	512
nilpotency class	3
isoclinism	isoclinic class of #250775
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

1132 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:

(2,7)(3,8)(4,5)(11,20)(13,14)(16,21)(18,28)(22,37)(23,24)(25,27)(29,39)(30,34)(31,32)(35,38)(36,40);(1,2,6,3)(4,8,5,7)(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);(1,3,6,2)(4,8,5,7)(9,11)(10,16)(12,22)(13,24)(14,20)(15,23)(17,29)(18,34)(19,30)(21,28)(25,35)(26,38)(27,37)(31,36)(32,39)(33,40);(1,2,6,3)(4,8,5,7)(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|b^2=h,c^2=h,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fhi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157994);