#6351065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60832	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2691 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,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, 6258261);