#145030

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

Polycyclic presentation:

2^9|b^2=i,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, 52226);