#179198

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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

721 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)(11,19,20,23)(12,15,13,17)(16,21,24,22)(27,33)(28,35)(30,38)(32,36)(34,37)(39,40);(1,2)(3,6)(4,8)(5,7)(10,15)(12,13)(16,23)(17,18)(19,24)(21,22)(25,26)(27,30)(28,36)(29,31)(32,35)(33,40)(34,39)(37,38);(3,4)(6,8)(9,11)(10,16)(12,22)(13,21)(14,20)(15,23)(17,19)(18,24)(25,27,29,34)(26,30,31,39)(28,37,35,33)(32,40,36,38);(1,3,5,4)(2,6,7,8)(11,20)(16,24)(19,23)(21,22)(26,31)(30,39)(32,36)(38,40)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=hi,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, 86394);