#445121

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
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×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1033 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)(9,10,15,16,14,20,13,19)(11,21,22,33,25,17,24,30)(12,26,27,18,28,31,29,34)(23,37,36,35,38,40,39,32);(10,16)(11,22)(12,27)(13,15)(17,21)(18,34)(19,20)(23,38)(24,25)(28,29)(32,40)(35,37);(2,5)(6,8)(9,11,13,24,14,25,15,22)(10,17,19,33,20,21,16,30)(12,23,29,39,28,38,27,36)(18,32,26,40,34,35,31,37);(1,3,4,7)(2,6,5,8)(9,12,14,28)(10,18,20,34)(11,23,25,38)(13,29,15,27)(16,31,19,26)(17,32,21,35)(22,36,24,39)(30,37,33,40)

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^d=ei,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, 352317);