#144366

order	512
nilpotency class	4
isoclinism	isoclinic class of #144366
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

2083 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 2083 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2)(3,6)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16);(2,6)(7,13)(8,14)(11,15);(3,9)(6,13)(10,16)(14,15)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51562);