#127305

order	512
nilpotency class	4
isoclinism	isoclinic class of #127265
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

419 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)(3,11,14,15)(4,8)(5,9)(6,10)(7,13,16,12)(17,18,22,26,23,27,20,24)(19,21,29,31,25,32,28,30);(1,3)(2,7,10,16)(4,12)(5,13)(6,14)(8,15,9,11)(17,19,21,24,22,28,31,27,23,25,32,26,20,29,30,18);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,23,22)(18,24,27,26)(19,28,25,29)(21,30,32,31)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=ghi,e^2=i,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ci,d^b=di,e^b=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34501);