#144882

order	512
nilpotency class	4
isoclinism	isoclinic class of #142392
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 426 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)(5,6,13,18)(7,11,21,30)(8,22,27,35)(9,17,12,15)(10,16,28,20)(14,32,34,25)(19,31,36,24)(23,29,33,26);(1,2)(3,4)(5,7,13,21)(6,10,18,28)(8,23,27,33)(9,16,12,20)(11,15,30,17)(14,24,34,31)(19,29,36,26)(22,32,35,25);(1,3,4,2)(5,8,9,24,13,27,12,31)(6,14,15,33,18,34,17,23)(7,19,20,22,21,36,16,35)(10,29,11,25,28,26,30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=i,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=egi,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, 52078);