#6443969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351165);