#178970

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

720 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)(9,10,14,18)(11,16,20,24)(12,17,13,15)(19,22,21,23)(25,26)(27,30)(28,31)(29,32)(33,39)(34,37)(35,38)(36,40);(10,15)(12,13)(16,22)(17,18)(19,21)(23,24)(25,27,29,34)(26,30,32,37)(28,33,35,40)(31,36,38,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,20)(10,16,18,24)(12,21,13,19)(15,22,17,23)(25,28,29,35)(26,31,32,38)(27,33,34,40)(30,36,37,39);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,22,24,23)(26,32)(27,34)(31,38)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86166);