C₅×#21684 #11944404

order	1280
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#21684

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₅×#2397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

1154 conjugacy classes of subgroups; only the 390 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 390 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) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 14672);