#142907

order	512
nilpotency class	4
isoclinism	isoclinic class of #142392
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	C₂×SD₁₆
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 432 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gi,f^2=gi,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=fgi,g^b=gi,d^c=dh,f^c=fg,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, 50103);