#143391

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6,11,15)(7,16,8,19)(9,14,10,13)(12,18,20,17)(21,22,27,33)(23,34,39,51)(24,40,44,49)(25,32,26,31)(28,35,47,36)(29,43,30,42)(37,48,38,50)(41,46,52,45);(1,2)(3,4)(5,7,11,8)(6,12,15,20)(9,18,10,17)(13,19,14,16)(21,23,27,39)(22,28,33,47)(24,41,44,52)(25,37,26,38)(29,48,30,50)(31,45,32,46)(34,49,51,40)(35,43,36,42);(2,4)(5,8,11,7)(6,12,15,20)(9,17,10,18)(13,19,14,16)(21,24,26,43,27,44,25,42)(22,29,32,49,33,30,31,40)(23,35,37,52,39,36,38,41)(28,34,45,50,47,51,46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,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=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=ehi

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, 50587);