#145965

order	512
nilpotency class	4
isoclinism	isoclinic class of #145951
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2065
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,c^b=cf,f^b=fi,d^c=dgi,e^c=eh,f^c=fg,h^c=hi

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, 53161);