#507667

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,36,34,43,33,42,22,27,21,26,20,25,38,47)(19,28,45,41,30,48,35,24,29,46,31,40,32,44,37,39);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(19,29)(24,39)(28,46)(30,32)(31,45)(35,37)(40,41)(44,48);(9,11,12,16)(10,14,13,15)(17,19,22,29)(18,24,27,39)(20,30,34,32)(21,35,23,37)(25,40,43,41)(26,44,36,48)(28,47,46,42)(31,38,45,33);(2,6)(5,7)(10,13)(14,15)(17,20,22,34)(18,25,27,43)(19,30,29,32)(21,33,23,38)(24,40,39,41)(26,42,36,47)(28,48,46,44)(31,37,45,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=g,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414863);