#128184

order	512
nilpotency class	4
isoclinism	isoclinic class of #128055
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fh,c^b=ch,d^b=dh,e^b=eg,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35380);