#250860

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

1126 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)(9,10,15,20)(11,16,23,32)(12,24,26,35)(13,19,14,18)(17,28,34,27)(21,33,37,40)(22,36,29,31)(25,30,38,39);(1,2,6,3)(4,8,5,7)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,36)(20,32)(21,28)(24,39)(26,38)(27,37)(30,35)(34,40);(1,3,6,2)(4,8,5,7)(9,12)(10,17)(11,21)(13,28)(14,27)(15,26)(16,30)(18,35)(19,24)(20,34)(22,38)(23,37)(25,29)(31,40)(32,39)(33,36);(1,2,6,3)(4,8,5,7)(9,11)(10,16)(12,21)(13,29)(14,22)(15,23)(17,30)(18,36)(19,31)(20,32)(24,40)(25,28)(26,37)(27,38)(33,35)(34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,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, 158056);