#153382

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,8)(4,6)(5,7)(9,10,12,17,15,19,25,14)(11,16,21,30,24,31,23,29)(13,18,26,32,28,35,34,27)(20,33,36,40,38,39,37,22);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,21)(13,20)(14,29)(15,24)(17,30)(18,33)(19,31)(22,27)(23,25)(26,36)(28,38)(32,40)(34,37)(35,39);(1,4,5,3)(2,6,7,8)(10,14)(11,20)(12,25)(13,27)(16,22)(17,19)(18,34)(21,37)(23,36)(24,38)(26,35)(28,32)(29,33)(30,39)(31,40)

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=d,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,h^b=hi,e^c=ei,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60578);