#138535

order	512
nilpotency class	4
isoclinism	isoclinic class of #138488
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18,23,28,30,27,39,26,24,29,38,40,22,19,21,32)(20,31,35,46,34,25,41,43,36,45,48,42,37,44,33,47);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(17,19,30,29,24,27,22,18)(20,31,34,44,36,45,37,25)(21,32,23,40,39,26,38,28)(33,46,35,47,41,42,48,43);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(17,20,24,36)(18,25,29,44)(19,31,27,45)(21,33,39,41)(22,37,30,34)(23,35,38,48)(26,42,32,46)(28,43,40,47)

Polycyclic presentation:

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

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, 45731);