#6449575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356771);