#144728

order	512
nilpotency class	4
isoclinism	isoclinic class of #144728
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10395
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#29751	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 423 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)(3,12)(4,6)(5,19)(9,24)(10,20)(11,17)(13,26)(14,21)(15,30)(16,31)(18,22)(25,32)(27,28);(1,2)(3,13)(4,17)(5,20)(6,8)(7,11)(9,15)(10,22)(12,24)(14,27)(16,25)(18,23)(19,29)(21,26)(28,31)(30,32);(1,3)(2,9,11,27)(4,15)(5,21)(6,14)(7,16)(8,13,17,25)(10,28,20,24)(12,19)(18,31)(22,30)(23,26,29,32)

Polycyclic presentation:

2^9|c^2=dg,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fi,d^c=di,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, 51924);