#179606

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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

737 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)(27,33)(28,35)(30,37)(32,39)(34,36)(38,40);(1,2)(3,6)(4,7)(5,8)(10,15)(12,13)(16,22)(17,18)(19,21)(23,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,38)(35,39)(36,40);(3,4)(6,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,23)(18,24)(25,27,29,34)(26,30,31,38)(28,36,35,33)(32,40,39,37);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,15,18,17)(11,19,20,21)(16,23,24,22)(26,31)(30,38)(32,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86802);