#431126

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

872 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)(3,7,8,5)(9,10,13,15)(11,16,12,14)(17,18)(19,27)(20,25)(21,26)(22,31)(23,28)(24,29)(30,32);(1,3,4,8)(2,5,6,7)(17,19,20,28)(18,23,25,27)(21,30,22,29)(24,31,32,26);(9,11,13,12)(10,14,15,16)(17,18,20,25)(19,24,28,32)(21,31,22,26)(23,29,27,30);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,32)(27,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=f,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,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, 338322);