#429351

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=f,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, 336547);