#6590175

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18,22,26)(19,27,30,39)(20,24,33,43)(21,34,37,25)(23,35,40,29)(28,45,46,41)(31,48,47,38)(32,44,36,42);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(17,19,22,30)(18,23,26,40)(20,31,33,47)(21,35,37,29)(24,41,43,45)(25,39,34,27)(28,36,46,32)(38,44,48,42);(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,28)(21,36)(22,33)(23,38)(25,44)(26,43)(27,45)(29,47)(30,46)(31,35)(32,37)(34,42)(39,41)(40,48);(1,4,5,3)(2,8,6,7)(18,25)(20,32)(21,37)(23,39)(26,34)(27,40)(28,31)(29,35)(33,36)(38,48)(42,44)(46,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497371);