#6368338

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
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

1526 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)(4,6)(5,14)(7,13)(8,15)(10,11)(12,16)(17,18,21,25)(19,26,28,31)(20,24,22,29)(23,30,32,27);(1,3)(2,7)(4,10)(5,11)(6,12)(8,16)(9,14)(13,15)(17,18)(19,23)(20,29)(21,25)(22,24)(26,27)(28,32)(30,31);(1,4,6,2)(3,10,12,7)(5,14,15,8)(9,13,16,11)(17,19)(18,23)(20,27)(21,28)(22,30)(24,31)(25,32)(26,29);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,32,26);(2,4)(3,12)(8,14)(11,13)(18,25)(23,32)(24,29)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275534);