#6589709

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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496905);