#6587648

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(12,13)(15,16)(17,18,21,24)(19,23,29,38)(20,25,33,39)(22,26,34,40)(27,37,30,35)(28,36,44,47)(31,42,32,41)(43,48,45,46);(2,6)(7,8)(17,19,21,29)(18,23,24,38)(20,28,33,44)(22,27,34,30)(25,36,39,47)(26,37,40,35)(31,45,32,43)(41,48,42,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(18,24)(19,27)(20,31)(22,34)(23,35)(25,41)(28,44)(29,30)(32,33)(37,38)(39,42)(46,48);(9,11,12,15)(10,14,13,16)(17,20,21,33)(18,25,24,39)(19,28,29,44)(22,32,34,31)(23,36,38,47)(26,41,40,42)(27,43,30,45)(35,46,37,48);(1,4,5,3)(2,8,6,7)(19,27)(20,32)(22,34)(23,37)(25,41)(26,40)(29,30)(31,33)(35,38)(39,42)(43,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,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, 6494844);