#135148

order	512
nilpotency class	4
isoclinism	isoclinic class of #135117
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

230 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=i,d^2=gi,e^2=ghi,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=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42344);