#6581448

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(11,13)(12,14)(15,16)(17,18,21,24)(19,23,29,38)(20,25,33,39)(22,26,34,40)(27,37,30,35)(28,36,44,47)(31,42,32,41)(43,48,45,46);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,38)(25,36)(26,37)(30,34)(31,45)(32,43)(33,44)(35,40)(39,47)(41,48)(42,46);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,24)(19,27)(20,31)(22,34)(23,35)(25,41)(28,44)(29,30)(32,33)(37,38)(39,42)(46,48);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(19,28)(21,33)(22,32)(23,36)(24,39)(26,41)(27,43)(29,44)(30,45)(31,34)(35,46)(37,48)(38,47)(40,42);(19,27)(20,32)(22,34)(23,37)(25,41)(26,40)(29,30)(31,33)(35,38)(39,42)(43,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=f,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, 6488644);