#148715

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

550 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=eh,f^c=fi,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, 55911);