#6356260

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356241
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
prime radical	#60836	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263456);