#149211

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

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=ei,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi

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, 56407);