#177728

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	#2682	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

720 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)(9,10,14,19)(11,16,21,24)(12,18,13,17)(15,22,23,20)(26,30)(28,35)(31,39)(32,36)(34,37)(38,40);(3,7)(5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,17)(18,24)(19,23)(25,26)(27,31)(28,36)(29,32)(30,35)(33,40)(34,38)(37,39);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,22)(13,20)(14,21)(15,17)(18,23)(19,24)(25,27,29,33)(26,31,32,40)(28,37,35,34)(30,38,36,39);(9,12,14,13)(10,17,19,18)(11,20,21,22)(15,16,23,24)(27,33)(31,40)(34,37)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,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, 84924);