#472245

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16815
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379441);