#144051

order	512
nilpotency class	4
isoclinism	isoclinic class of #143959
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33580	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,d^c=dgi,e^c=eh,f^c=fg,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, 51247);