#146220

order	512
nilpotency class	4
isoclinism	isoclinic class of #146219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 265 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=de,c^2=i,d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fhi,g^c=gi

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, 53416);