#6443963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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

1990 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)(11,15)(12,25)(13,18)(14,19)(16,27)(17,30)(20,37)(21,23)(22,32)(24,33)(26,31)(28,35)(29,36)(34,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,26)(13,23)(14,22)(16,34)(17,33)(18,21)(19,32)(20,28)(24,30)(25,31)(27,38)(29,39)(35,37)(36,40);(9,12,13,27,14,29,30,28)(10,16,17,25,19,35,18,36)(11,20,23,26,22,38,24,39)(15,31,33,37,32,40,21,34);(1,4,5,3)(2,6,7,8)(9,10)(11,21)(12,16)(13,17)(14,19)(15,23)(18,30)(20,34)(22,33)(24,32)(25,27)(26,31)(28,36)(29,35)(37,38)(39,40);(11,22)(15,32)(20,38)(21,33)(23,24)(26,39)(31,40)(34,37)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6351159);