#6589721

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(17,18,23,22)(19,24,28,31)(20,25,21,26)(27,30,32,29);(2,6)(7,8)(17,19)(18,24)(20,29)(21,30)(22,31)(23,28)(25,27)(26,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,27)(21,23)(22,26)(24,30)(28,32)(29,31);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(18,22)(19,28)(20,25)(21,26)(27,30)(29,32);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(19,27)(20,23)(22,25)(24,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=fi,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=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, 6496917);