#6588989

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(12,13)(15,16)(17,18,22,27)(19,29,31,24)(20,32,34,25)(21,26,37,42)(23,28,38,41)(30,40,47,44)(33,43,36,48)(35,46,39,45);(1,3,5,4)(2,6,8,7)(17,19,22,31)(18,24,27,29)(20,33,34,36)(21,23,37,38)(25,43,32,48)(26,41,42,28)(30,39,47,35)(40,46,44,45);(10,13)(14,16)(17,20,22,34)(18,25,27,32)(19,30,31,47)(21,36,37,33)(23,39,38,35)(24,40,29,44)(26,43,42,48)(28,46,41,45);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,23)(20,30)(22,37)(24,41)(25,44)(27,42)(28,29)(31,38)(32,40)(33,39)(34,47)(35,36)(43,45)(46,48);(1,4,5,3)(2,6,8,7)(19,21)(23,38)(24,42)(26,29)(28,41)(30,36)(31,37)(33,47)(35,39)(40,48)(43,44)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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, 6496185);