#143221

order	512
nilpotency class	4
isoclinism	isoclinic class of #143219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28497	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

521 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,6)(4,7)(5,8)(9,10,15,21)(11,17,24,35)(12,16,28,32)(13,20,14,19)(18,27,31,25)(22,37,29,34)(23,39,26,30)(33,40,36,38);(2,6)(3,4)(7,8)(10,16)(11,22)(12,25)(13,14)(17,33)(18,20)(19,31)(21,32)(23,26)(24,29)(27,28)(30,34)(35,36)(37,39);(2,7)(3,4)(6,8)(9,11)(10,17)(12,26)(13,29)(14,22)(15,24)(16,30)(18,36)(19,37)(20,34)(21,35)(23,28)(25,40)(27,38)(31,33)(32,39)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=ehi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50417);