#445126

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

985 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)(3,6)(4,5)(7,8)(9,10,13,19,15,16,14,27)(11,17,20,33,23,29,22,31)(12,24,25,18,26,34,28,30)(21,36,35,32,37,40,38,39);(10,16)(11,20)(13,14)(17,31)(21,35)(22,23)(24,34)(25,28)(29,33)(32,40)(36,39)(37,38);(2,5)(6,8)(9,11,13,20,15,23,14,22)(10,17,19,33,16,29,27,31)(12,21,25,35,26,37,28,38)(18,32,34,40,30,39,24,36);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,25)(14,28)(15,26)(16,30)(17,32)(19,34)(20,35)(22,38)(23,37)(24,27)(29,39)(31,36)(33,40)

Polycyclic presentation:

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

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, 352322);