#462772

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

731 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=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, 369968);