#356526

order	512
nilpotency class	3
isoclinism	isoclinic class of #356478
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13809
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×H(3,ℤ/4) ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,g^b=gi,f^c=fi,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263722);