#143334

order	512
nilpotency class	4
isoclinism	isoclinic class of #143209
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	#28497	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

460 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,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, 50530);