#6356567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356533
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61999	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1635 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)(6,16)(7,10)(12,14)(13,15)(17,18,22,24)(19,21,26,31)(20,27,30,25)(23,28,32,29);(1,3,6,8,7,13,5,12)(2,4,9,11,10,14,16,15)(17,19,21,24,22,26,31,18)(20,25,28,32,30,27,29,23);(17,20)(18,23)(19,25)(21,28)(22,30)(24,32)(26,27)(29,31);(1,4,7,14)(2,8,10,12)(3,9,13,16)(5,15,6,11);(1,5,7,6)(2,9,10,16)(3,12,13,8)(4,11,14,15)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340125, #72340127, #72340133, …
maximal normal subgroup of	#72340125, #72340127, #72340133, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263763);