#125752

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, prime radical	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
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

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

normal subgroups only

This group has 355 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) = 512

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,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, 32948);