#6439752

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#23891	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 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)(3,6)(4,5)(7,8)(9,10,16,21)(11,17,24,12)(13,19,30,37)(14,31,15,20)(18,27,23,26)(22,33,39,25)(28,34,38,40)(29,36,32,35);(2,5)(6,8)(9,11)(10,17)(12,21)(13,28)(14,23)(15,18)(16,24)(19,34)(20,27)(22,29)(25,35)(26,31)(30,38)(32,39)(33,36)(37,40);(9,12,16,17)(10,18,21,23)(11,20,24,31)(13,25,30,33)(14,27,15,26)(19,28,37,38)(22,35,39,36)(29,34,32,40);(9,13)(10,19)(11,22)(12,25)(14,29)(15,32)(16,30)(17,33)(18,28)(20,35)(21,37)(23,38)(24,39)(26,40)(27,34)(31,36);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346948);