#6742705

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,34)(21,26)(22,27)(23,28)(25,37)(29,45)(30,33)(31,44)(32,42)(35,48)(36,38)(39,40)(41,47)(43,46);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19,22,32)(18,24,27,42)(20,35,37,46)(21,30,28,44)(23,31,26,33)(25,43,34,48)(29,40,47,38)(36,45,39,41);(10,14)(13,15)(17,20,22,37)(18,25,27,34)(19,29,32,47)(21,39,28,36)(23,40,26,38)(24,41,42,45)(30,48,44,43)(31,35,33,46);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,22,28)(18,26,27,23)(19,30,32,44)(20,36,37,39)(24,33,42,31)(25,40,34,38)(29,43,47,48)(35,45,46,41);(1,5,6,4)(2,7,3,8)(11,16)(13,15)(18,27)(19,31)(20,37)(23,26)(24,30)(29,46)(32,33)(35,47)(36,39)(41,43)(42,44)(45,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fg,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, 6649901);