#6841940

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	#42602	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(19,28,29,24)(20,25,31,21)(23,27,32,30);(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,24,26,28)(20,30,31,27)(21,23,25,32);(2,6)(7,8)(10,14)(11,16)(17,20,23,24,22,31,32,28)(18,25,27,19,26,21,30,29);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,26)(20,31)(24,28)(27,30);(1,4,5,3)(2,8,6,7)(17,21,32,19,22,25,23,29)(18,20,30,28,26,31,27,24)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=fgi,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, 6749136);