#142904

order	512
nilpotency class	4
isoclinism	isoclinic class of #142394
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,24,55)(8,14,28,48)(9,29,10,34)(11,38,32,17)(13,40,44,21)(15,37,16,33)(20,54,58,67)(22,49,23,51)(25,60,62,65)(26,50,63,52)(27,53,39,47)(30,42,35,43)(31,46,36,64)(41,61,66,59)(45,56,68,57);(5,7,11,23,12,24,32,22)(6,13,17,43,18,44,38,42)(8,25,27,59,28,62,39,61)(9,30,33,40,10,35,37,21)(14,45,47,54,48,68,53,67)(15,49,29,55,16,51,34,19)(20,36,57,26,58,31,56,63)(41,52,60,46,66,50,65,64);(2,4)(5,8,12,28)(6,14,18,48)(7,20,24,58)(9,31,10,36)(11,39,32,27)(13,41,44,66)(15,50,16,52)(17,53,38,47)(19,54,55,67)(21,59,40,61)(22,57,23,56)(25,35,62,30)(26,37,63,33)(29,46,34,64)(42,60,43,65)(45,51,68,49)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fg,g^b=gi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50100);