#6359336

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359226
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1591 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)(3,9)(4,13)(5,7)(6,12)(8,14)(10,16)(11,15)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(1,3,5,11)(2,6,7,16)(4,10,14,12)(8,15,13,9)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12);(2,7)(3,11)(8,13)(10,12)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340116, #72340122, #72340134
maximal normal subgroup of	#72340116, #72340122, #72340134

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266532);