#143294

order	512
nilpotency class	4
isoclinism	isoclinic class of #143211
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	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 397 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=eh

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, 50490);