#6362098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362022
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

1533 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)(17,18,22,24)(19,26,27,32)(20,25,21,29)(23,30,31,28);(1,3)(2,7)(4,10)(5,11)(6,12)(8,16)(9,14)(13,15)(19,27)(23,31)(26,32)(28,30);(1,4,6,2)(3,10,12,7)(5,14,15,8)(9,13,16,11)(17,19)(18,23)(20,28)(21,30)(22,27)(24,31)(25,32)(26,29);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)(18,24)(23,31)(25,29)(26,32);(2,4)(3,12)(8,14)(11,13)(17,20,22,21)(18,25,24,29)(19,28,27,30)(23,32,31,26)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,e^c=eh,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, 6269294);