#138978

order	512
nilpotency class	4
isoclinism	isoclinic class of #138838
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 ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 358 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)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(17,18,22,26,38,47,21,39,24,30,20,27,23,29,40,28)(19,31,34,45,41,42,37,25,36,44,32,48,35,43,33,46);(1,3,5,10)(2,6)(4,9,11,15)(7,12)(8,13)(14,16)(17,19)(18,25)(20,37)(21,32)(22,33)(23,41)(24,36)(26,42)(27,43)(28,44)(29,48)(30,46)(31,39)(34,40)(35,38)(45,47);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(18,26)(20,21)(22,40)(23,38)(25,42)(27,30)(28,47)(29,39)(31,48)(32,37)(33,34)(35,41)(43,46)(44,45)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=g,d^2=ghi,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46174);