#137496

order	512
nilpotency class	4
isoclinism	isoclinic class of #137493
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=gi,e^2=ghi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44692);