#6840322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(4,8)(5,6)(9,10,13,15)(11,16,12,14)(17,18,22,26)(19,28,29,24)(20,25,31,21)(23,27,32,30);(9,10)(11,14)(12,16)(13,15)(17,19,22,29)(18,24,26,28)(20,30,31,27)(21,23,25,32);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,23,24,22,31,32,28)(18,25,27,19,26,21,30,29);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(18,26)(20,31)(24,28)(27,30);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,32,19,22,25,23,29)(18,20,30,28,26,31,27,24)

Polycyclic presentation:

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

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, 6747518);