#6579642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(19,26)(23,31)(27,28)(30,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18,21,25)(19,23,26,31)(20,22,29,24)(27,32,28,30);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,22)(19,27)(20,29)(23,31)(24,25)(26,28);(17,19,21,26)(18,23,25,31)(20,28,29,27)(22,30,24,32);(18,22)(19,28)(20,29)(24,25)(26,27)(30,32)

Polycyclic presentation:

2^9|b^2=fgi,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,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, 6486838);