#6361617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6361570
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

1532 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6268813);