#6364181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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

1528 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,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, 6271377);