#6357902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
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	#3511	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1831 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)(19,25)(21,22)(26,27)(28,29)(31,32);(17,18,22,27,23,26,21,24)(19,20,25,32,29,30,28,31);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,22,25,23,29,21,28)(18,20,27,32,26,30,24,31);(3,10)(6,15)(9,16)(11,14)(17,20)(18,25)(19,24)(21,31)(22,32)(23,30)(26,28)(27,29);(18,26)(20,30)(24,27)(31,32)

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,e^b=eh,d^c=di,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, 6265098);