#139123

order	512
nilpotency class	4
isoclinism	isoclinic class of #139083
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=eh,f^c=fghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46319);