#126097

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

277 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(5,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26,29,33)(27,35,36,53)(28,39,40,55)(30,44,42,34)(31,46,47,37)(32,49,50,43)(38,54,45,51)(41,52,56,48);(1,2,4,6)(3,5,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(28,40)(32,50)(35,53)(37,46)(39,55)(41,56)(43,49)(51,54);(1,3,4,7)(2,5,6,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,27)(26,31)(28,41)(29,36)(30,45)(32,51)(33,47)(34,52)(35,39)(37,43)(38,42)(40,56)(44,48)(46,49)(50,54)(53,55)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fgi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,h^a=hi,d^b=df,e^b=eg,d^c=dg,e^c=eh,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, 33293);