#6357328

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	#3499	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=g,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, 6264524);