#6357783

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357777
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1948 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)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16);(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(19,26)(22,31)(25,32)(27,30);(3,10)(6,15)(9,16)(11,14)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306699, #72306743, #72306763
maximal normal subgroup of	#72306699, #72306743, #72306763

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264979);