#6355987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355925
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1626 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(19,27)(20,21)(23,30)(25,26)(28,29)(31,32);(1,3,4,12,6,11,5,13)(2,7,14,16,9,15,8,10)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(3,11)(7,15)(10,16)(12,13)(17,19,22,28)(18,23,24,31)(20,27,21,29)(25,30,26,32);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,15)(12,16)(18,24)(23,31)(25,26)(30,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20,22,21)(18,25,24,26)(19,27,28,29)(23,30,31,32)

Polycyclic presentation:

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

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, 6263183);