#170123

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

741 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77319);