#6742061

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

1554 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6649257);