#142465

order	512
nilpotency class	4
isoclinism	isoclinic class of #142224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,22,27)(19,28,31,41)(20,32,35,25)(21,24,36,26)(23,33,38,29)(30,43,46,44)(34,42,48,40)(37,45,39,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,22,31)(18,23,27,38)(20,33,35,29)(21,30,36,46)(24,39,26,37)(25,41,32,28)(34,47,48,45)(40,44,42,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(21,36)(23,37)(25,40)(26,27)(28,43)(29,33)(30,46)(32,42)(34,48)(38,39)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49661);