#149350

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#476
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

528 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=de,d^2=h,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ehi,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,g^c=gh,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, 56546);