#136151

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1862
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43347);