#6590661

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	#36019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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)(17,18,21,25)(19,27,31,23)(20,24,33,40)(22,26,35,42)(28,38,30,36)(29,43,45,37)(32,39,34,41)(44,48,46,47);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,29)(21,31)(22,28)(24,37)(25,27)(26,36)(30,35)(32,44)(33,45)(34,46)(38,42)(39,47)(40,43)(41,48);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(19,28)(20,32)(22,35)(23,36)(24,39)(26,42)(27,38)(30,31)(33,34)(40,41)(44,46)(47,48);(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,29)(21,33)(22,32)(23,37)(25,40)(26,39)(27,43)(28,44)(30,46)(31,45)(34,35)(36,47)(38,48)(41,42);(1,4,5,3)(2,6,8,7)(19,30)(20,32)(22,35)(23,38)(24,39)(26,42)(27,36)(28,31)(29,45)(33,34)(37,43)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fi,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,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, 6497857);