#179224

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	#45962	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

757 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)(11,19,12,16)(13,22,23,17)(15,21,24,20)(25,26,30,34)(27,35,36,39)(28,31,29,33)(32,37,40,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,12)(10,15,18,24)(13,21,23,20)(16,22,19,17)(26,31)(28,29)(32,39)(33,34)(35,40)(37,38);(3,4)(6,8)(9,12,14,11)(10,16,18,19)(13,21,23,20)(15,22,24,17)(25,27)(26,32)(28,38)(29,37)(30,36)(31,39)(33,35)(34,40);(1,3,5,4)(2,6,7,8)(11,12)(15,24)(16,19)(20,21)(27,36)(32,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 86420);