#250650

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

1112 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)(11,21)(12,26)(13,19)(14,17)(15,20)(16,22)(18,28)(23,31)(24,33)(25,32)(27,35)(29,36)(30,34)(37,39)(38,40);(1,2)(3,5)(4,7)(6,8)(9,11)(10,16)(12,27)(13,24)(14,22)(15,25)(17,32)(18,35)(19,21)(20,33)(23,30)(26,40)(28,39)(29,38)(31,36)(34,37);(10,17)(11,22)(13,14)(18,34)(19,20)(21,32)(23,38)(24,25)(26,36)(27,39)(28,29)(37,40);(1,3,5,2)(4,6,8,7)(9,12)(10,18)(11,23)(13,29)(14,28)(15,30)(16,31)(17,34)(19,26)(20,36)(21,37)(22,38)(24,39)(25,27)(32,40)(33,35)

Polycyclic presentation:

2^9|d^2=hi,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, 157846);