#6589595

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1571 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,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, 6496791);