#172568

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

815 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)(9,10,15,12)(11,18,22,24)(13,17,14,23)(16,20,19,21)(25,26,30,34)(27,35,36,39)(28,31,29,33)(32,37,40,38);(1,2,4,6)(3,5,7,8)(9,11,15,22)(10,16,12,19)(13,21,14,20)(17,24,23,18)(26,31)(28,29)(32,39)(33,34)(35,40)(37,38);(1,3)(2,5)(4,7)(6,8)(9,12,15,10)(11,19,22,16)(13,17,14,23)(18,21,24,20)(25,27,30,36)(26,32,34,40)(28,38,29,37)(31,39,33,35);(2,6)(5,8)(9,13,15,14)(10,17,12,23)(11,20,22,21)(16,18,19,24)(27,36)(32,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,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, 79764);