#143893

order	512
nilpotency class	4
isoclinism	isoclinic class of #143821
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	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 329 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,d^c=dgi,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51089);