#170492

order	512
nilpotency class	3
isoclinism	isoclinic class of #170381
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

733 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77688);