#135800

order	512
nilpotency class	4
isoclinism	isoclinic class of #135766
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#499
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

181 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 94 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)(17,18,22,28,23,26,45,29)(19,30,33,55,34,24,35,48)(20,36,39,67,40,66,41,73)(21,42,43,27,44,54,72,53)(25,49,52,38,51,70,56,71)(31,58,60,79,37,68,65,80)(32,61,62,47,63,74,64,57)(46,75,76,69,50,78,77,59);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(17,19,22,35,23,34,45,33)(18,24,28,55,26,30,29,48)(20,37,41,65,40,31,39,60)(21,32,43,64,44,63,72,62)(25,50,56,77,51,46,52,76)(27,47,54,61,53,57,42,74)(36,58,73,79,66,68,67,80)(38,69,49,78,71,59,70,75);(17,20,21,38,22,41,43,49,23,40,44,71,45,39,72,70)(18,25,27,36,29,52,42,67,26,51,53,66,28,56,54,73)(19,31,32,59,35,60,64,75,34,37,63,69,33,65,62,78)(24,46,47,68,48,77,74,79,30,50,57,58,55,76,61,80)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42996);