#6681827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6589023);