#6442023

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,22)(19,25)(20,27)(24,28)(26,29)(30,32);(10,14)(13,16)(17,18,21,24)(19,23,26,31)(20,28,27,22)(25,30,29,32);(17,19,21,26)(18,23,24,31)(20,29,27,25)(22,30,28,32);(1,3,4,7)(2,6,5,8)(19,26)(23,31)(25,29)(30,32)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,h^c=hi,e^d=ei,h^d=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, 6349219);