#136203

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

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

normal subgroups only

This group has 356 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,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, 43399);