#144576

order	512
nilpotency class	4
isoclinism	isoclinic class of #144565
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=i,e^2=ghi,f^2=g,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51772);