#444800

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=g,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,h^a=hi,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, 351996);