#6446231

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,32)(22,26)(23,27)(28,29)(30,31);(1,3,5,4)(2,6,8,7)(17,18)(19,28)(20,31)(21,26)(22,32)(23,27)(24,29)(25,30);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,23,30)(18,24,27,31)(20,21,29,22)(25,32,28,26);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(19,21)(22,30)(23,29)(24,32)(26,31)(27,28);(17,21,23,22)(18,26,27,32)(19,29,30,20)(24,25,31,28)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=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, 6353427);