#474391

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381587);