#144879

order	512
nilpotency class	4
isoclinism	isoclinic class of #142256
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

448 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,e^c=eg,f^c=fgi,g^c=gi,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, 52075);