#6443736

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,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, 6350932);