C₄⋊#338 #10838

order	256
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

381 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(2,4)(5,6,8,12,10,13,9,18)(7,14,16,11,17,19,15,20);(5,7,9,16,10,17,8,15)(6,11,12,14,13,20,18,19);(1,2,3,4)

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=e,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,g^a=gh,f^b=fh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3826);