#6590689

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 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)(12,16)(13,15)(17,18,23,20)(19,29,32,25)(21,26,36,34)(22,27,24,28)(30,33,41,38)(31,43,46,40)(35,39,42,37)(44,47,48,45);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,29)(21,31)(22,33)(23,32)(24,38)(26,40)(27,30)(28,41)(34,43)(35,44)(36,46)(37,47)(39,45)(42,48);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,23,18)(19,30,32,41)(21,35,36,42)(22,27,24,28)(25,38,29,33)(26,39,34,37)(31,43,46,40)(44,45,48,47);(9,12,13,11)(10,16,15,14)(17,21)(18,26)(19,31)(20,34)(22,37)(23,36)(24,39)(25,40)(27,35)(28,42)(29,43)(30,44)(32,46)(33,47)(38,45)(41,48);(1,4,5,3)(2,6,8,7)(17,22)(18,27)(19,32)(20,28)(23,24)(25,29)(31,45)(35,42)(37,39)(40,48)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,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, 6497885);