#167661

order	512
nilpotency class	3
isoclinism	isoclinic class of #166993
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×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

745 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)(9,10,14,13)(11,15,19,18)(12,16,21,20)(17,23,24,22)(25,26,29,32)(27,33,34,40)(28,36,37,31)(30,38,39,35);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(28,37)(31,36)(33,40)(35,38);(1,3,5,4)(2,6,8,7)(9,12)(10,16)(11,17)(13,20)(14,21)(15,22)(18,23)(19,24)(25,27)(26,30)(28,38)(29,34)(31,40)(32,39)(33,36)(35,37);(1,3,5,4)(2,7,8,6)(9,13,14,10)(11,18,19,15)(12,20,21,16)(17,23,24,22)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=ghi,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, 74857);