#469733

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16661
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

948 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)(3,13,9,8)(4,17,19,16)(5,10)(7,14,22,12)(15,23)(18,30)(20,28)(21,26)(24,27)(25,32)(29,31);(1,3)(2,8)(4,14)(5,15)(6,9)(7,17)(10,23)(11,13)(12,19)(16,22)(18,28)(20,30)(21,24)(25,31)(26,27)(29,32);(1,4,7,13,6,19,22,8)(2,9,12,17,11,3,14,16)(5,18,20,27,21,31,32,23)(10,24,25,30,26,15,28,29);(1,5,6,21)(2,10)(3,15,9,24)(4,18)(7,20,22,32)(8,23)(11,26)(12,25)(13,27)(14,28)(16,29,17,30)(19,31)

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei,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, 376929);