#143706

order	512
nilpotency class	4
isoclinism	isoclinic class of #143705
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19187	C₂
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

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

normal subgroups only

This group has 472 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) = 128

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

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

Polycyclic presentation:

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

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, 50902);