#145482

order	512
nilpotency class	4
isoclinism	isoclinic class of #145482
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33779	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,5)(4,7)(6,8)(10,16)(11,13)(12,21)(14,20)(17,24)(18,22)(19,23);(3,8)(5,6)(9,10)(11,17)(12,22)(13,16)(14,24)(15,19)(18,20)(21,23);(2,5)(3,8)(6,7)(9,10,11,16,20,24,21,22,15,19,12,23,14,18,13,17)

Polycyclic presentation:

2^9|c^2=d,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,e^c=egh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52678);