#6349912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349901
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

2907 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(18,23)(19,26)(20,21)(25,29)(27,28)(30,31);(4,13)(8,12)(9,16)(11,14)(17,18)(19,24)(20,23)(21,29)(22,25)(26,30)(27,31)(28,32);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,19)(18,24)(20,26)(21,27)(22,28)(23,30)(25,32)(29,31);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,23,25,29)(19,26,28,27)(24,30,32,31);(17,20,22,21)(18,23,25,29)(19,27,28,26)(24,31,32,30)

Polycyclic presentation:

2^9|d^2=h,e^2=h,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,e^c=eh,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257658, #72306700, #72306703, …
maximal normal subgroup of	#72257658, #72306700, #72306703, …
maximal quotient of	#72264340

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257108);