#138519

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	#29748	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 422 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,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, 45715);