#141804

order	512
nilpotency class	4
isoclinism	isoclinic class of #141746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 78 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)(9,10,14,21,15,19,16,38)(11,17,25,41,23,42,36,43)(12,26,29,59,30,58,37,60)(13,31,33,49,34,66,35,20)(18,44,47,28,46,63,48,64)(22,51,54,71,52,72,27,57)(24,55,56,40,32,65,68,50)(39,61,45,53,69,62,70,67);(2,6,8,7)(9,11,16,25,15,23,14,36)(10,17,21,43,19,42,38,41)(12,27,37,22,30,54,29,52)(13,32,35,68,34,24,33,56)(18,45,47,39,46,70,48,69)(20,50,31,65,49,40,66,55)(26,57,59,72,58,71,60,51)(28,62,44,67,64,61,63,53);(9,12,13,28,16,29,35,63,15,30,34,64,14,37,33,44)(10,18,20,26,21,48,31,60,19,46,49,58,38,47,66,59)(11,22,24,53,25,27,56,61,23,52,32,67,36,54,68,62)(17,39,40,51,43,45,55,71,42,69,50,72,41,70,65,57)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=e,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ch,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, 49000);