#141738

order	512
nilpotency class	4
isoclinism	isoclinic class of #141698
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,7)(4,8)(5,6)(9,10)(11,23)(12,30,34,60)(13,36)(14,19)(15,40)(16,22)(17,26)(18,33,47,54)(20,38)(21,37)(24,49)(25,41,57,70)(27,43)(28,50)(29,53)(31,61,63,45)(32,64,66,48)(35,68,65,46)(39,52)(42,58)(44,59)(51,56,71,62)(55,69,67,72);(1,3,7,2,5,8,4,6)(9,11,24,27,15,28,13,26,16,29,14,39,37,58,38,59)(10,17,20,50,21,43,19,23,22,44,36,42,40,52,49,53)(12,31,33,67,35,57,66,56,34,63,54,55,65,25,32,62)(18,45,60,71,48,70,68,72,47,61,30,51,64,41,46,69);(2,6)(3,8)(9,12,24,54,15,35,13,32,16,34,14,33,37,65,38,66)(10,18,36,60,21,48,49,68,22,47,20,30,40,64,19,46)(11,25,39,62,28,31,59,67,29,57,27,56,58,63,26,55)(17,41,50,72,43,45,23,51,44,70,42,69,52,61,53,71)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,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, 48934);