#144694

order	512
nilpotency class	4
isoclinism	isoclinic class of #144694
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10394
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

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

normal subgroups only

This group has 419 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)(5,18)(9,25)(10,19)(11,16)(13,31)(14,20)(15,30)(17,21)(22,29)(23,26)(27,28);(1,2)(3,13)(4,16)(5,19)(6,8)(7,11)(9,15)(10,21)(12,28)(14,23)(17,24)(18,32)(20,31)(22,27)(25,29)(26,30);(1,3,4,14,7,15,6,22)(2,9,8,23,11,13,16,27)(5,20,17,12,21,29,18,30)(10,25,24,28,19,31,32,26)

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=di,e^b=eh,e^c=egi,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, 51890);