#136706

order	512
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

376 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 112 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)(5,6,8,11,9,7,10,12)(13,14,18,23,19,24,17,35,20,25,16,32,34,39,33,21)(15,26,29,22,30,38,28,40,31,43,27,41,37,44,36,42);(1,2,3,4)(5,7,8,12,9,6,10,11)(14,21,25,35)(16,33)(17,18)(19,34)(22,38,41,44)(23,24,32,39)(26,42,43,40)(27,36)(28,29)(30,37);(13,15,20,31)(14,22,25,41)(16,28,18,36)(17,27,33,29)(19,37,34,30)(21,38,35,44)(23,26,32,43)(24,42,39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43902);