#6348389

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	#62805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2834 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,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	#72271563, #72271566, #72271575
maximal normal subgroup of	#72271563, #72271566, #72271575

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255585);