#179704

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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

800 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)(9,10,14,18)(11,19,22,23)(12,17,13,16)(15,20,24,21)(26,31)(27,34)(28,29)(33,37)(35,36)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,19,17,23)(25,26,30,33)(27,32,36,40)(28,31,29,37)(34,38,35,39);(1,3,4,7)(2,5,6,8)(12,13)(16,17)(19,23)(20,21)(25,27,30,36)(26,32,33,40)(28,34,29,35)(31,38,37,39);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)(25,28,30,29)(26,31,33,37)(27,35,36,34)(32,39,40,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,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, 86900);