#143756

order	512
nilpotency class	4
isoclinism	isoclinic class of #143711
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50952);