#142921

order	512
nilpotency class	4
isoclinism	isoclinic class of #142232
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi

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, 50117);