#144756

order	512
nilpotency class	4
isoclinism	isoclinic class of #144756
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

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

normal subgroups only

This group has 350 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=d,e^2=gi,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,f^b=fi,e^c=eg,f^c=fi,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, 51952);