#167516

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

859 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)(25,26,30,33)(27,35,36,39)(28,34,29,32)(31,38,40,37);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,19,17,23)(25,27,30,36)(26,31,33,40)(28,38,29,37)(32,39,34,35);(1,3)(2,5)(4,7)(6,8)(26,32)(28,29)(31,39)(33,34)(35,40)(37,38);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)(26,33)(27,36)(32,34)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,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, 74712);