#143437

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 ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 380 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,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, 50633);