#137376

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

597 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44572);