#474131

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381327);