#440648

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 347844);