#6354235

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354207
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

1653 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,27)(20,25)(21,24)(22,26)(23,28)(29,31)(30,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,15)(12,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(3,11)(7,15)(10,16)(12,13)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,32,31);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,31,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306619, #72306633, #72306697, …
maximal normal subgroup of	#72306619, #72306633, #72306697, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261431);