#144027

order	512
nilpotency class	4
isoclinism	isoclinic class of #143935
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

357 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51223);