#386945

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

896 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)(11,21,25,37)(12,26,23,40)(13,29)(14,19)(15,20)(16,24,34,27)(17,28,36,30)(18,31)(22,35)(32,38)(33,39);(1,3,5,2)(4,7,8,6)(9,11,15,25)(10,16,20,34)(12,22,23,39)(13,30,31,28)(14,27,32,24)(17,35,36,33)(18,26,29,40)(19,37,38,21);(1,4,5,8)(2,6,3,7)(9,12)(10,17,20,36)(11,22)(13,27)(14,28)(15,23)(16,33,34,35)(18,21,29,37)(19,26,38,40)(24,31)(25,39)(30,32);(2,3)(6,7)(9,13,14,22,15,31,32,39)(10,18,19,33,20,29,38,35)(11,23,24,30,25,12,27,28)(16,17,21,26,34,36,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294141);