#426023

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15345
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333219);