#134618

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

314 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=g,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41814);