#6362157

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362118
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	#61496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1577 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,23)(20,21)(24,31)(25,29)(26,32)(27,30);(17,18)(19,26)(20,29)(21,23)(22,25)(24,27)(28,31)(30,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,18)(19,24)(20,23)(21,29)(22,25)(26,27)(28,32)(30,31);(17,19)(18,24)(20,30)(21,27)(22,28)(23,31)(25,32)(26,29);(2,7)(6,15)(8,12)(9,16)

Polycyclic presentation:

2^9|c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,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, 6269353);