#250401

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
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

1095 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)(9,10,14,18)(11,15,21,29)(12,16,22,30)(13,24,25,17)(19,32,26,27)(20,34,35,28)(23,37,36,31)(33,39,38,40);(1,2)(3,8)(4,6)(5,7)(11,19)(13,25)(15,27)(17,24)(20,33)(21,26)(23,36)(28,39)(29,32)(31,37)(34,40)(35,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,18,29)(12,20,22,35)(13,26,25,19)(16,28,30,34)(17,32,24,27)(23,38,36,33)(31,40,37,39);(1,2)(3,8)(4,6)(5,7)(9,12,14,22)(10,16,18,30)(11,20,21,35)(13,23,25,36)(15,28,29,34)(17,31,24,37)(19,33,26,38)(27,39,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157597);