#6367128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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

1662 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,e^2=hi,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,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, 6274324);