#141910

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49106);