#6590178

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

1742 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)(19,23,30,38)(20,24,32,40)(21,25,33,41)(27,35,29,37)(28,36,44,47)(31,39,34,42)(43,46,45,48);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,27)(22,30)(24,36)(25,35)(26,38)(29,33)(31,45)(32,44)(34,43)(37,41)(39,48)(40,47)(42,46);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(19,27)(20,31)(21,33)(23,35)(24,39)(25,41)(28,44)(29,30)(32,34)(36,47)(37,38)(40,42);(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,28)(21,34)(22,32)(23,36)(25,42)(26,40)(27,43)(29,45)(30,44)(31,33)(35,46)(37,48)(38,47)(39,41);(1,4,5,3)(2,8,6,7)(19,29)(20,31)(21,33)(23,37)(24,39)(25,41)(27,30)(32,34)(35,38)(40,42)(43,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,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, 6497374);