#144797

order	512
nilpotency class	4
isoclinism	isoclinic class of #142308
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51993);