#444457

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351653);