#143749

order	512
nilpotency class	4
isoclinism	isoclinic class of #143701
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28240	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,f^b=fi,e^c=eh,f^c=fi,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, 50945);