#433437

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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)(3,5)(4,6)(7,8)(9,10,14,30,15,20,28,19)(11,21,31,33,25,38,24,16)(12,17,23,35,27,32,26,36)(13,18,22,39,29,34,40,37);(1,3,4,7)(2,5,6,8)(9,11,14,31,15,25,28,24)(10,16,30,21,20,33,19,38)(12,22,23,29,27,40,26,13)(17,34,35,37,32,18,36,39);(2,6)(5,8)(9,12,14,26,15,27,28,23)(10,17,19,35,20,32,30,36)(11,22,24,29,25,40,31,13)(16,18,21,37,33,34,38,39);(9,13,28,40,15,29,14,22)(10,18,30,39,20,34,19,37)(11,23,31,27,25,26,24,12)(16,32,38,35,33,17,21,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340633);