#370369

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fg,d^2=e,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, 277565);