C₅×#29685 #11952405

order	1280
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#29685

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₅×#3459	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

1176 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1176 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) = 320

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

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

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, 22673);