#6481609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470019
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	C₂²×C₄	C₂⁵
prime radical	#60805	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2926 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,26)(20,23)(21,25)(22,30)(24,28)(27,31)(29,32);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,24)(20,27)(21,28)(22,31)(23,29)(25,32)(26,30);(3,9)(6,13)(10,11)(14,15)(17,20,23,18)(19,27,29,24)(21,30,22,25)(26,31,32,28);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26);(2,8)(6,13)(7,16)(14,15)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388805);