#6348440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348364
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
prime radical	#62815	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306813, #72306887, #72306924
maximal normal subgroup of	#72306813, #72306887, #72306924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255636);