#145370

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

336 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,h^b=hi,e^c=egh

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, 52566);