#142926

order	512
nilpotency class	4
isoclinism	isoclinic class of #142828
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fg,g^c=gi,h^c=hi

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, 50122);