#6580018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573477
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	#32904	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,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, 6487214);