#116266

order	512
nilpotency class	3
isoclinism	isoclinic class of #9938
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

711 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)(3,6)(4,7)(5,8)(9,10,13,17,14,18,25,32)(11,15,21,28,20,27,34,38)(12,16,24,31,23,30,36,40)(19,29,33,39,22,26,35,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,22,23,19)(13,21)(14,20)(16,29,30,26)(17,28)(18,27)(24,35,36,33)(25,34)(31,39,40,37)(32,38);(2,6)(3,4)(7,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,31)(18,30)(20,22)(21,33)(25,36)(27,29)(28,37)(32,40)(34,35)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23462);