#6349116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349099
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

2923 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:

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

Polycyclic presentation:

2^9|d^2=i,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,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72272561, #72272568, #72272666, …
maximal normal subgroup of	#72272561, #72272568, #72272666, …
maximal quotient of	#72341054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256312);