#6590985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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)(11,14)(12,15)(13,16)(17,18,22,26)(19,23,30,39)(20,24,34,43)(21,25,35,44)(27,36,29,38)(28,37,31,40)(32,41,33,42)(45,47,46,48);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,31)(21,27)(22,30)(24,40)(25,36)(26,39)(28,34)(29,35)(32,45)(33,46)(37,43)(38,44)(41,47)(42,48);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,16,14)(19,27)(20,32)(21,35)(23,36)(24,41)(25,44)(28,31)(29,30)(33,34)(37,40)(38,39)(42,43);(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,28)(21,33)(22,34)(23,37)(25,42)(26,43)(27,45)(29,46)(30,31)(32,35)(36,47)(38,48)(39,40)(41,44);(1,4,5,3)(2,6,8,7)(19,27)(20,33)(21,35)(23,36)(24,42)(25,44)(29,30)(32,34)(38,39)(41,43)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498181);