#6681995

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(26,31)(28,39)(32,45)(34,43)(35,48)(36,40)(41,53)(52,55);(1,3,4,7)(2,5,6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,26,29,31)(27,35,37,48)(28,33,39,46)(30,43,42,34)(32,44,45,38)(36,51,40,54)(41,56,53,50)(47,55,49,52);(2,6)(5,8)(25,27,29,37)(26,32,31,45)(28,40,39,36)(30,44,42,38)(33,49,46,47)(34,48,43,35)(41,52,53,55)(50,51,56,54);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,39)(26,33,31,46)(27,36,37,40)(30,41,42,53)(32,47,45,49)(34,50,43,56)(35,51,48,54)(38,52,44,55);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(26,34)(30,42)(31,43)(32,48)(33,50)(35,45)(38,44)(41,53)(46,56)(47,54)(49,51)(52,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=f,e^2=fi,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, 6589191);