#138416

order	512
nilpotency class	4
isoclinism	isoclinic class of #138415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45612);