#125751

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1939 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

371 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=fi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,h^a=hi,d^b=df,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32947);