#6356727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356633
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

1605 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263923);