#6359137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359112
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

1703 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266333);