#172679

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

696 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,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, 79875);