#142860

order	512
nilpotency class	4
isoclinism	isoclinic class of #142801
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#2619	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fgi,h^b=hi,d^c=dh,e^c=ei,f^c=fgi,f^d=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, 50056);