#179491

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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	#7742	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

805 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)(10,15)(12,13)(16,22)(17,18)(19,21)(23,24)(30,36)(37,40);(1,2,6,3)(4,8,5,7)(9,10,14,18)(11,16,20,23)(12,15,13,17)(19,24,21,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,36)(35,38)(39,40);(1,3,6,2)(4,8,5,7)(9,11,14,20)(10,16,18,23)(12,21,13,19)(15,22,17,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,36)(35,39)(38,40);(1,4,6,5)(2,8,3,7)(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,22,23,24)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,37,36,40)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,d^b=dgi,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, 86687);