#444396

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

883 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,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, 351592);