#6366652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
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

1579 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)(3,7)(4,10)(5,14)(6,9)(8,11)(12,15)(13,16)(17,18)(19,29)(20,25)(21,30)(22,27)(23,26)(24,28)(31,32);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,24,31)(18,20,28,21)(22,30,23,25)(26,32,27,29);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20,24,21)(18,19,28,31)(22,29,23,32)(25,27,30,26);(2,9)(7,15)(8,16)(10,14)(17,21,24,20)(18,25,28,30)(19,27,31,26)(22,29,23,32);(1,4,6,5)(2,10,9,14)(3,11,12,13)(7,8,15,16)(17,22,24,23)(18,26,28,27)(19,30,31,25)(20,32,21,29)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283802, #72284002
maximal normal subgroup of	#72283802, #72284002

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273848);