#172832

order	512
nilpotency class	3
isoclinism	isoclinic class of #172777
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

781 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 80028);