#250123

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	#61097	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1209 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:

(1,2,4,7)(3,5,8,6)(13,25)(17,26)(19,32)(20,34)(22,27)(24,36)(28,39)(31,37);(2,5)(3,8)(6,7)(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);(9,11,14,21)(10,15,18,29)(12,20,23,34)(13,27,25,22)(16,28,30,39)(17,32,26,19)(24,38,36,35)(31,40,37,33);(2,5)(3,8)(6,7)(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|a^2=h,b^2=i,c^2=i,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, 157319);