#6349131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349099
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

2876 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:

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

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306531, #72306696, #72306740, …
maximal normal subgroup of	#72306531, #72306696, #72306740, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256327);