#6355530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1763 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339685, #72339700, #72339800, …
maximal normal subgroup of	#72339685, #72339700, #72339800, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262726);