#172628

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

772 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79824);