#6820783

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	#33179	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ghi,d^2=i,e^2=fhi,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, 6727979);