#145016

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

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

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

normal subgroups only

This group has 394 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)(4,7)(6,8)(9,10)(11,15)(12,24)(13,22)(14,26)(16,20)(17,23)(18,25)(19,29)(21,33)(27,35)(28,38)(30,37)(31,39)(32,40)(34,36);(9,11,17,26)(10,18,23,33)(12,27,32,36)(13,21,16,25)(14,22,15,20)(19,28,37,31)(24,29,40,30)(34,38,35,39);(2,5)(3,8)(6,7)(9,12,13,28,17,32,16,31)(10,19,20,36,23,37,22,27)(11,24,25,39,26,40,21,38)(14,34,18,29,15,35,33,30)

Polycyclic presentation:

2^9|b^2=i,c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=egi,f^c=fh,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, 52212);