#6841834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,30,32)(20,27,31,24)(21,29,28,25);(9,11)(10,13)(12,16)(14,15)(17,18)(19,28)(20,24)(21,30)(22,26)(23,29)(25,32)(27,31);(2,6)(7,8)(10,14)(11,16)(17,19,20,29,22,30,31,25)(18,23,27,28,26,32,24,21);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,22,31)(18,24,26,27)(19,29,30,25)(21,32,28,23);(1,4,5,3)(2,8,6,7)(17,21,31,32,22,28,20,23)(18,25,24,30,26,29,27,19)

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=fg,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6749030);