#444007

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

878 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)(9,10,14,19)(11,20,23,29)(12,24,27,17)(13,15,28,18)(16,31,33,21)(22,35,38,40)(25,30,26,34)(32,37,39,36);(1,3,4,8)(2,5,6,7)(10,15)(11,21)(12,25)(13,28)(17,24)(18,19)(20,29)(23,31)(26,27)(32,40)(35,39)(36,37);(9,11,13,21,14,23,28,31)(10,16,18,20,19,33,15,29)(12,22,26,36,27,38,25,37)(17,32,34,35,24,39,30,40);(3,8)(5,7)(9,12)(10,17)(11,22)(13,26)(14,27)(15,30)(16,32)(18,34)(19,24)(20,35)(21,36)(23,38)(25,28)(29,40)(31,37)(33,39)

Polycyclic presentation:

2^9|a^2=ef,b^2=efi,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,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, 351203);