#250712

order	512
nilpotency class	3
isoclinism	isoclinic class of #250647
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

1102 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,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,d^b=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157908);