#6348468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348447
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2730 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,15)(14,16)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,2)(3,7)(4,9)(5,8)(6,10)(11,12)(13,16)(14,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,11,16,15)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306718, #72306720, #72306722, …
maximal normal subgroup of	#72306718, #72306720, #72306722, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255664);