#166016

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
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

835 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 835 conjugacy classes of subgroups — too many to draw. Only its 142 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)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24)(25,26,29,33)(27,30,35,39)(28,31,36,32)(34,37,40,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18)(25,27,29,35)(26,30,33,39)(28,34,36,40)(31,37,32,38);(1,3)(2,5)(4,7)(6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39);(10,17)(15,24)(16,21)(18,23)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,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, 73212);