#126612

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,f^a=fg,e^b=eg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33808);