#144754

order	512
nilpotency class	4
isoclinism	isoclinic class of #144754
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10396
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51950);