#19745

order	256
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

497 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

2^8|b^2=e,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,d^c=dh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12733);