#168752

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

920 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75948);