#166138

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1089 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1089 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,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,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, 73334);