#6367196

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

1595 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)(17,18)(19,29)(20,24)(21,26)(22,28)(23,27)(25,31)(30,32);(1,2,6,10)(3,7,5,9)(4,12,13,15)(8,14,16,11)(17,19,22,20)(18,24,28,29)(21,31,23,30)(25,26,32,27);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,20,22,19)(18,25,28,32)(21,31,23,30)(24,26,29,27);(1,4,6,13)(2,8,10,16)(3,11,5,14)(7,15,9,12)(17,21,22,23)(18,26,28,27)(19,30,20,31)(24,25,29,32);(1,3,6,5)(2,9,10,7)(4,11,13,14)(8,12,16,15)(17,21,22,23)(18,27,28,26)(19,30,20,31)(24,32,29,25)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=hi,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, 6274392);