#250089

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
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
prime radical	#62529	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157285);