#143317

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
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 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

436 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,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, 50513);