#168121

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	#37129	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1061 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1061 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,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, 75317);