#138484

order	512
nilpotency class	4
isoclinism	isoclinic class of #138482
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,h^b=hi,d^c=dg,f^c=fg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45680);