#6590304

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 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)(17,18,22,26)(19,23,30,39)(20,31,34,24)(21,25,35,42)(27,36,29,38)(28,44,45,37)(32,41,33,40)(43,48,46,47);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,23,26,39)(20,28,34,45)(21,27,35,29)(24,37,31,44)(25,36,42,38)(32,46,33,43)(40,48,41,47);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(19,27)(20,32)(21,35)(23,36)(24,40)(25,42)(28,45)(29,30)(31,41)(33,34)(37,44)(38,39);(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,28)(21,33)(22,34)(23,37)(25,41)(26,31)(27,43)(29,46)(30,45)(32,35)(36,47)(38,48)(39,44)(40,42);(1,4,5,3)(2,8,6,7)(19,27)(20,33)(21,35)(23,36)(24,41)(25,42)(29,30)(31,40)(32,34)(38,39)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 6497500);