#138481

order	512
nilpotency class	4
isoclinism	isoclinic class of #138421
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=g,d^2=ghi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45677);