#433436

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=e,d^2=e,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, 340632);