#6345878

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61450	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dh,f^a=fh,f^b=fh,e^d=ehi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253074);