#6359853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359664
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

1553 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6267049);