#433569

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340765);