#168828

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 770 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=h,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, 76024);