#6679011

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6586207);