#6349904

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

3290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|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	#72257636, #72257658
maximal normal subgroup of	#72257636, #72257658
maximal quotient of	#72264142

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257100);