#6444128

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	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,24)(13,28)(14,18)(15,19)(17,27)(20,34)(21,29)(22,32)(23,33)(25,39)(26,36)(30,35)(31,38)(37,40);(2,6)(7,8)(9,11,15,23)(10,16,19,33)(12,25,26,38)(13,22,14,29)(17,34,35,40)(18,21,28,32)(20,30,37,27)(24,39,36,31);(1,3,5,4)(2,7,6,8)(9,12,13,27,15,26,14,30)(10,17,18,24,19,35,28,36)(11,20,22,38,23,37,29,25)(16,31,21,34,33,39,32,40);(1,4,5,3)(2,8,6,7)(9,10)(11,21)(12,17)(13,18)(14,28)(15,19)(16,29)(20,34)(22,33)(23,32)(24,27)(25,31)(26,35)(30,36)(37,40)(38,39);(12,26)(17,35)(20,37)(24,36)(25,38)(27,30)(31,39)(34,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^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,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, 6351324);