#180255

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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

734 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24)(25,26,31,36)(27,37,28,40)(29,34,30,35)(32,38,33,39);(1,2,5,8)(3,6,4,7)(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18)(25,27,31,28)(26,32,36,33)(29,38,30,39)(34,37,35,40);(3,4)(6,7)(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,24)(21,23)(25,28,31,27)(26,33,36,32)(29,38,30,39)(34,37,35,40);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(25,29,31,30)(26,34,36,35)(27,38,28,39)(32,37,33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87451);