#6357330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357317
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3498	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1917 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264526);