#444003

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

864 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=g,d^2=i,e^2=i,f^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,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, 351199);