#141828

order	512
nilpotency class	4
isoclinism	isoclinic class of #141707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19667	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 307 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=g,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49024);