C₅×#12132 #11934852

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12090
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅
prime radical	C₅×#1595	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₂²⋊C₄ ◁ G

544 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15)(16,18)(17,19)(20,21);(6,7)(8,11)(9,13)(10,12)(15,18)(19,21);(6,8,10,9)(7,11,12,13)(16,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5120);