#6587762

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(19,23)(20,24)(21,26)(22,25)(27,35)(28,36)(29,37)(30,38)(31,39)(32,42)(33,40)(34,41)(43,46)(44,47)(45,48);(2,6)(7,8)(17,19,22,30)(18,23,25,38)(20,28,33,44)(21,27,34,29)(24,36,40,47)(26,35,41,37)(31,45,32,43)(39,48,42,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(19,27)(20,31)(21,34)(23,35)(24,39)(26,41)(28,44)(29,30)(32,33)(36,47)(37,38)(40,42);(9,11,12,15)(10,14,13,16)(17,20,22,33)(18,24,25,40)(19,28,30,44)(21,32,34,31)(23,36,38,47)(26,42,41,39)(27,43,29,45)(35,46,37,48);(1,4,5,3)(2,8,6,7)(18,25)(19,27)(20,32)(21,34)(23,37)(24,39)(29,30)(31,33)(35,38)(36,47)(40,42)(43,45)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6494958);