#143565

order	512
nilpotency class	4
isoclinism	isoclinic class of #143516
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9)(3,10,13,15)(4,7,14,8)(6,11,16,12)(17,18)(19,27)(20,31)(21,36)(22,25)(23,28)(24,34)(26,33)(29,44)(30,47)(32,42)(35,43)(37,40)(38,46)(39,48)(41,45);(1,3)(2,6)(4,11)(5,13)(7,10)(8,15)(9,16)(12,14)(17,19,22,29)(18,23,25,38)(20,32,35,48)(21,37,33,30)(24,40,41,47)(26,42,36,39)(27,43,44,31)(28,45,46,34);(2,7)(3,11)(4,14)(8,9)(10,15)(12,13)(23,38)(27,44)(39,42)(40,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50761);