#6726570

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1269 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)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,28)(20,33)(21,37)(22,27)(23,26)(24,31)(25,34)(29,40)(30,39)(32,38)(35,41)(36,46)(42,45)(43,47)(44,48);(2,5)(3,7)(9,11,13,12)(10,14,16,15)(17,19,22,31)(18,24,27,28)(20,30,34,45)(21,32,23,29)(25,39,33,42)(26,38,37,40)(35,44,36,43)(41,47,46,48);(2,5)(6,8)(9,11,13,12)(10,15,16,14)(19,29)(20,34)(21,23)(24,38)(25,33)(26,37)(28,40)(30,44)(31,32)(39,48)(42,47)(43,45);(2,5)(6,8)(17,20,22,34)(18,25,27,33)(19,30,31,45)(21,35,23,36)(24,39,28,42)(26,41,37,46)(29,43,32,44)(38,47,40,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,31)(20,35)(22,23)(25,41)(27,37)(30,45)(33,46)(34,36)(38,40)(47,48)

Polycyclic presentation:

2^9|b^2=fg,c^2=fgi,d^2=i,e^2=g,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, 6633766);