#144555

order	512
nilpotency class	4
isoclinism	isoclinic class of #143205
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 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dhi,c^2=hi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fh,e^c=eg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51751);