#6589726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#33187	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,13)(15,16)(19,27)(20,23)(24,29)(28,30);(2,6)(7,8)(17,18,21,26)(19,28,27,30)(20,29,23,24)(22,32,31,25);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,21,27)(18,23,26,20)(22,24,31,29)(25,30,32,28);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,23)(18,24,26,29)(19,25,27,32)(22,28,31,30);(1,4,5,3)(2,8,6,7)(18,25)(20,28)(22,31)(23,30)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,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=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496922);