C₅×#11532 #11934252

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11532

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅
prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×Q₈ ◁ G

412 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 29 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4520);