#6367608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
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

1511 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,24)(19,26)(21,22)(25,32)(27,28)(29,31);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,19)(20,28,30,32)(21,26,22,24)(25,31,27,29);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,18)(20,27,30,25)(21,26,22,24)(28,29,32,31);(17,20,23,30)(18,25,19,27)(21,31,22,29)(24,32,26,28);(2,7)(6,15)(8,12)(9,16)(18,19)(24,26)(25,27)(28,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^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,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274804);