#6357026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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	#61988	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1545 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264222);