#250553

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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	#61080	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1225 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)(9,10,14,19)(11,20,24,31)(12,25,28,33)(13,16,29,18)(15,21,32,23)(17,27,34,26)(22,30,38,40)(35,37,39,36);(2,5)(3,8)(6,7)(9,11)(10,15)(12,22)(13,21)(14,24)(16,20)(17,30)(18,31)(19,32)(23,29)(25,35)(26,37)(27,36)(28,38)(33,39)(34,40);(10,16)(11,21)(13,29)(17,25)(18,19)(20,31)(22,36)(23,24)(26,27)(33,34)(35,39)(37,38);(2,5)(3,8)(6,7)(9,12,14,28)(10,17,19,34)(11,22,24,38)(13,27,29,26)(15,30,32,40)(16,25,18,33)(20,35,31,39)(21,36,23,37)

Polycyclic presentation:

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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157749);