#142291

order	512
nilpotency class	4
isoclinism	isoclinic class of #142244
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	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

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

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, 49487);