#141789

order	512
nilpotency class	4
isoclinism	isoclinic class of #141653
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

306 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,4,7,5,8,3,6)(9,10,14,21,15,19,16,36)(11,17,25,39,23,40,35,41)(12,26,29,55,30,53,31,59)(13,32,33,45,34,60,27,20)(18,42,44,57,43,56,54,28)(22,46,50,69,48,70,58,71)(24,52,47,38,51,66,61,67)(37,62,63,72,64,68,65,49);(2,6,8,7)(9,11,16,25,15,23,14,35)(10,17,21,41,19,40,36,39)(12,22,31,50,30,48,29,58)(13,24,27,47,34,51,33,61)(18,37,44,65,43,64,54,63)(20,38,32,52,45,67,60,66)(26,46,55,71,53,70,59,69)(28,49,56,62,57,72,42,68);(9,12,27,56,16,31,34,57,15,30,33,42,14,29,13,28)(10,18,32,55,21,44,45,53,19,43,60,59,36,54,20,26)(11,22,47,62,25,50,51,72,23,48,61,68,35,58,24,49)(17,37,52,71,41,65,67,70,40,64,66,69,39,63,38,46)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48985);