#6580434

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	#24450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1694 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,29)(22,26)(23,27)(28,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,29)(21,25)(22,28)(23,30)(26,32)(27,31);(10,14)(13,16)(19,28)(20,31)(21,22)(24,32)(25,30)(26,29);(1,4,5,3)(2,6,7,8)(17,20,23,32)(18,25,27,28)(19,29,30,26)(21,31,22,24);(17,21)(18,26)(22,23)(24,31)(25,28)(27,29)

Polycyclic presentation:

2^9|b^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,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, 6487630);