#145121

order	512
nilpotency class	4
isoclinism	isoclinic class of #143329
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1634
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10430
prime radical	#28249	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52317);