#177690

order	512
nilpotency class	3
isoclinism	isoclinic class of #177385
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	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

713 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)(3,7)(6,8)(10,14)(12,19)(15,23)(16,20)(18,21)(22,24)(25,26,30,34)(27,35,38,39)(28,33,29,32)(31,36,40,37);(3,7)(5,8)(9,10,13,16)(11,15,17,24)(12,20,19,14)(18,22,21,23)(25,27,30,38)(26,31,34,40)(28,36,29,37)(32,39,33,35);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,21)(13,17)(14,22)(16,24)(18,19)(20,23)(26,32)(28,29)(31,39)(33,34)(35,40)(36,37);(11,17)(15,24)(18,21)(22,23)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,40,35)

Polycyclic presentation:

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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 84886);