#6355982

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

1587 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,e^2=i,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, 6263178);