#169066

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76262);