C₅×#7602 #11930322

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7602

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

408 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 1280

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 590);