#6741683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,45)(29,38)(31,48)(32,41)(33,42)(34,43)(35,39)(36,44)(37,46)(40,47);(1,3)(2,5)(4,8)(6,7)(17,19,21,30)(18,23,25,27)(20,31,32,47)(22,35,34,29)(24,40,41,48)(26,38,43,39)(28,36,46,33)(37,44,45,42);(2,6)(5,7)(10,14)(15,16)(17,20,21,32)(18,24,25,41)(19,28,30,46)(22,36,34,33)(23,37,27,45)(26,44,43,42)(29,47,35,31)(38,40,39,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,25)(20,32)(23,27)(26,43)(28,46)(31,47)(33,36)(38,39);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,25)(19,29)(20,32)(22,34)(23,38)(27,39)(28,31)(30,35)(37,48)(40,45)(42,44)(46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fgi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,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, 6648879);