#149326

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1982
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,h^b=hi,d^c=dh,e^c=ehi,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, 56522);