#134545

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

442 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dghi,c^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,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, 41741);