#140785

order	512
nilpotency class	4
isoclinism	isoclinic class of #135932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#442
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cgh,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47981);