#6365536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
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

1596 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18)(19,27)(20,28)(21,31)(22,25)(23,29)(24,30)(26,32);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,19,22,23)(18,20,25,30)(21,24,32,28)(26,29,31,27);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,22,30)(18,23,25,19)(21,27,32,29)(24,31,28,26);(2,8)(6,14)(7,9)(13,15)(17,20,22,30)(18,24,25,28)(19,26,23,31)(21,29,32,27);(3,10)(6,14)(11,16)(13,15)(18,25)(19,23)(24,28)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272732);