#6349062

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349016
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

2899 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)(4,8)(5,9)(6,10)(7,13)(12,15)(14,16)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,3)(2,7)(4,13)(5,12)(6,14)(8,16)(9,11)(10,15)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30);(17,19,22,27)(18,23,25,32)(20,26,21,29)(24,31,28,30);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,11,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339685, #72339691, #72339802
maximal normal subgroup of	#72339685, #72339691, #72339802

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256258);