#146268

order	512
nilpotency class	4
isoclinism	isoclinic class of #146219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

233 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=deg,c^2=g,d^2=gi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fh,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, 53464);