#143750

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	#532	D₄
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

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

normal subgroups only

This group has 398 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,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, 50946);