#134619

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

324 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,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, 41815);