#6456448

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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)(9,10,15,16,14,20,13,19)(11,21,27,31,22,17,24,29)(12,18,28,30,25,34,26,33)(23,35,38,40,36,32,37,39);(10,16)(11,22)(12,25)(13,15)(17,31)(18,33)(19,20)(21,29)(23,36)(30,34)(32,40)(35,39);(2,5)(6,8)(9,11,13,24,14,22,15,27)(10,17,19,31,20,21,16,29)(12,23,26,37,25,36,28,38)(18,32,33,40,34,35,30,39);(9,12,13,26,14,25,15,28)(10,18,19,33,20,34,16,30)(11,23,24,37,22,36,27,38)(17,32,31,40,21,35,29,39);(1,3,4,7)(2,6,5,8)(11,22)(17,21)(23,36)(24,27)(29,31)(32,35)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6363644);