#128109

order	512
nilpotency class	4
isoclinism	isoclinic class of #128001
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

718 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35305);