#138974

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

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

normal subgroups only

This group has 335 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)(17,18,22,29,43,44,41,27,23,30,42,28,24,31,21,40)(19,32,35,25,39,45,33,48,36,47,34,46,37,26,20,38);(1,3,5,10)(2,6)(4,9,11,15)(7,12)(8,13)(14,16)(17,19,24,39,23,36,43,37)(18,25,44,38,30,46,31,48)(20,21,35,42,33,41,34,22)(26,28,47,40,45,29,32,27);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20,23,33)(18,26,30,45)(19,21,36,41)(22,39,42,37)(24,35,43,34)(25,28,46,29)(27,38,40,48)(31,32,44,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,e^2=gh,f^2=gh,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, 46170);