#177846

order	512
nilpotency class	3
isoclinism	isoclinic class of #177733
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

883 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 883 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dgh,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, 85042);