#147670

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1092 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)(5,6,7,8)(9,10)(11,17)(12,16)(13,19)(14,28)(15,20)(18,24)(21,33)(22,29)(23,35)(25,30)(26,31)(27,34)(32,36)(37,40)(38,39);(10,16)(11,21)(12,24)(13,14)(17,32)(18,19)(20,31)(22,37)(23,27)(25,26)(28,30)(29,33)(34,40)(35,39);(2,4)(9,11)(10,17)(12,22)(13,27)(14,21)(15,23)(16,29)(18,32)(19,34)(20,35)(24,36)(25,38)(26,37)(28,33)(30,39)(31,40)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,f^b=fi,h^b=hi,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, 54866);