#152805

order	512
nilpotency class	5
isoclinism	isoclinic class of #151115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

176 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,6)(4,5)(7,8)(9,10,11,16,15,21,14,18)(12,24,22,32,25,17,28,34)(13,29,23,38,31,36,30,20)(19,37,33,26,39,40,35,27);(2,5)(6,8)(10,16,21,18)(12,25)(17,32,24,34)(19,35,39,33)(20,29,38,36)(22,28)(26,27)(37,40);(1,3,4,7)(2,6,5,8)(10,17)(11,14)(12,26)(13,30)(16,32)(18,34)(19,38)(20,39)(21,24)(22,37)(23,31)(25,27)(28,40)(29,33)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=dh,d^2=i,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60001);