#167782

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

783 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

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, 74978);