#144920

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	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 419 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=dg,d^2=i,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=di,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, 52116);