#168172

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
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

789 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75368);