#178422

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
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	#46287	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

855 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 855 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,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, 85618);