#6580428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#24428	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 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)(17,18,23,22)(19,21,25,27)(20,24,30,32)(26,31,29,28);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,23,25)(18,21,22,27)(20,28,30,31)(24,26,32,29);(10,14)(13,16)(17,20,23,30)(18,24,22,32)(19,26,25,29)(21,31,27,28);(1,4,5,3)(2,6,7,8)(17,21)(18,25)(19,22)(20,29)(23,27)(24,28)(26,30)(31,32);(18,22)(19,27)(21,25)(24,32)(26,28)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487624);