#153415

order	512
nilpotency class	5
isoclinism	isoclinic class of #153255
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

167 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=deh,c^2=d,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,e^b=ei,f^b=fg,g^b=gh,h^b=hi,f^c=fgi,g^c=gh,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, 60611);