#6362153

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1630 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=h,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, 6269349);