#6349929

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

2884 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(2,7)(4,5)(8,15)(9,10)(11,16)(13,14)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30);(1,3)(2,8)(4,14)(5,13)(6,12)(7,15)(9,16)(10,11)(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(2,9)(7,10)(8,16)(11,15)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30);(2,9)(3,12)(7,10)(13,14)

Polycyclic presentation:

2^9|d^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,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	#72306700, #72306703, #72306709, …
maximal normal subgroup of	#72306700, #72306703, #72306709, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257125);