#6358900

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266096);