#6524893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
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	#62880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2967 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)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,21,23,20,24,27,28)(19,22,26,30,25,29,31,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(21,27)(23,28)(26,31)(30,32);(2,7)(3,9)(4,11)(12,16)(17,18)(19,22)(20,24)(21,28)(23,27)(25,29)(26,32)(30,31);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(17,19,20,25)(18,22,24,29)(21,26,27,31)(23,30,28,32);(3,9)(6,13)(10,15)(12,16)(19,25)(22,29)(26,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6432089);