#370370

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

788 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,23,28,21)(13,29)(15,19)(16,20)(17,27,24,26)(18,30)(22,32)(25,37,40,33)(31,34)(35,38,36,39);(2,5)(6,7)(9,11,15,20,16,19,14,10)(12,24,26,21,28,17,27,23)(13,22,31,34,32,29,30,18)(25,37,38,35,40,33,39,36);(9,12,16,28)(10,17)(11,21)(13,25,32,40)(14,26,15,27)(18,33)(19,23)(20,24)(22,35)(29,36)(30,38,31,39)(34,37);(1,3,4,8)(2,6,5,7)(9,13,15,31,16,32,14,30)(10,18,11,22,20,34,19,29)(12,25,27,39,28,40,26,38)(17,33,21,35,24,37,23,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277566);