#136559

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1862
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

266 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=eg,d^c=dg,e^c=eh,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, 43755);