#6724850

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#42945	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,16,15)(17,18,23,26)(19,27,30,31)(20,25,22,32)(21,29,24,28);(9,11,13,12)(10,15,16,14)(17,19,23,30)(18,24,26,21)(20,29,22,28)(25,31,32,27);(2,6)(5,7)(17,20,23,22)(18,25,26,32)(19,28,30,29)(21,31,24,27);(3,8)(5,7)(9,12,13,11)(10,15,16,14)(17,21)(18,19)(20,31)(22,27)(23,24)(25,28)(26,30)(29,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6632046);