#135056

order	512
nilpotency class	4
isoclinism	isoclinic class of #134985
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#478
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18,22,28,23,26,44,29)(19,30,33,57,34,24,35,49)(20,36,39,55,37,70,40,73)(21,41,45,27,46,56,65,53)(25,50,51,71,52,43,58,38)(31,60,63,79,61,76,69,80)(32,64,67,48,42,74,68,54)(47,75,77,62,59,66,78,72);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(17,19,22,35,23,34,44,33)(18,24,28,57,26,30,29,49)(20,31,39,69,37,61,40,63)(21,42,45,67,46,32,65,68)(25,47,51,78,52,59,58,77)(27,54,56,74,53,48,41,64)(36,60,55,80,70,76,73,79)(38,72,50,66,71,62,43,75);(17,20,23,37)(18,25,26,52)(19,31,34,61)(21,43,46,50)(22,39,44,40)(24,47,30,59)(27,55,53,73)(28,51,29,58)(32,66,42,75)(33,63,35,69)(36,56,70,41)(38,65,71,45)(48,79,54,80)(49,77,57,78)(60,74,76,64)(62,67,72,68)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=i,d^2=gi,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42252);