#138985

order	512
nilpotency class	4
isoclinism	isoclinic class of #138844
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(17,18,22,29,40,43,38,25,24,31,20,27,23,30,21,28)(19,32,35,46,41,44,34,45,37,47,33,26,36,42,39,48);(1,3,5,10)(2,6)(4,9,11,15)(7,12)(8,13)(14,16)(18,25)(20,38)(21,22)(23,40)(26,42)(27,30)(28,31)(29,43)(32,45)(33,34)(35,39)(36,41)(44,46)(47,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,19,24,37)(18,26,31,46)(20,39,22,34)(21,33,38,35)(23,41,40,36)(25,42,28,44)(27,47,29,32)(30,48,43,45)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=gi,d^2=ghi,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=eghi,f^c=fghi,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46181);