#149213

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
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 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,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,d^c=di,f^c=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, 56409);