#180263

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

629 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32)(33,34,36,39)(35,41,43,48)(37,45,44,40)(38,46,47,42);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31)(33,34)(35,38)(36,39)(37,45)(40,44)(41,46)(42,48)(43,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(17,20,21,27)(18,23,24,30)(19,25,26,32)(22,28,29,31)(33,35)(34,38)(36,43)(37,46)(39,47)(40,48)(41,45)(42,44);(2,8)(3,9)(6,12)(10,15)(18,24)(22,29)(23,30)(28,31)(34,39)(38,47)(40,45)(41,48)

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87459);