#477920

order	512
nilpotency class	3
isoclinism	isoclinic class of #477920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17126
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385116);