#139026

order	512
nilpotency class	4
isoclinism	isoclinic class of #139008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#2184	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=ghi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=egh,f^c=fh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46222);