C₅×#29656 #11952376

order	1280
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#29656

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

1428 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 160

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

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 22644);