#145160

order	512
nilpotency class	4
isoclinism	isoclinic class of #145143
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28225	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

398 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=dgh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gi,c^b=cf,f^b=fh,d^c=di,e^c=egi,f^c=fh,g^c=gi

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, 52356);