#144861

order	512
nilpotency class	4
isoclinism	isoclinic class of #144674
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=dg,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fi,e^c=egi,h^c=hi,f^e=fi

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, 52057);