#143650

order	512
nilpotency class	4
isoclinism	isoclinic class of #143626
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	#33811	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

362 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 256

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

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

Polycyclic presentation:

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

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, 50846);