#6580133

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1498 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,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, 6487329);