#6586719

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=fi,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, 6493915);