#143378

order	512
nilpotency class	4
isoclinism	isoclinic class of #143217
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50574);