#145106

order	512
nilpotency class	4
isoclinism	isoclinic class of #145015
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=dg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52302);