#167441

order	512
nilpotency class	3
isoclinism	isoclinic class of #166993
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

893 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)(11,17)(12,20)(13,15)(14,19)(16,22)(18,24)(21,23)(25,26,30,33)(27,31,29,37)(28,35,36,39)(32,38,40,34);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22)(25,27,30,29)(26,31,33,37)(28,34,36,38)(32,39,40,35);(1,3)(2,5)(4,7)(6,8)(25,28,30,36)(26,32,33,40)(27,34,29,38)(31,39,37,35);(10,15)(11,18)(16,20)(19,21)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 74637);