#141678

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48874);