#142724

order	512
nilpotency class	4
isoclinism	isoclinic class of #142644
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	#33752	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 68 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)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,32)(20,37)(21,26)(22,30)(23,29)(24,31)(25,34)(27,39)(28,40)(33,42)(35,47)(36,46)(38,43)(41,44)(45,48);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,24,36)(18,25,31,45)(20,38,21,41)(22,33,23,35)(26,46,37,32)(27,47,28,42)(29,44,30,43)(34,40,48,39);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(18,26)(19,33)(20,39)(21,40)(22,23)(25,42)(27,29)(28,30)(31,37)(32,43)(34,48)(35,36)(44,46)(45,47)

Polycyclic presentation:

2^9|b^2=gh,d^2=gi,e^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49920);