#6360229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360176
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3480	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1541 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:

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

Polycyclic presentation:

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

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, 6267425);