#143488

order	512
nilpotency class	4
isoclinism	isoclinic class of #143452
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33779	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=gi,e^2=gh,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50684);