C₅×#33761 #11956481

order	1280
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#33761

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

584 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,19,23)(16,24,25,42)(17,28,32,43)(18,33,30,22)(20,35,36,26)(21,38,40,31)(27,44,34,39)(29,41,45,37);(6,8)(7,10)(9,13)(11,12)(16,25)(20,36)(24,42)(26,35)(27,34)(29,45)(37,41)(39,44);(8,13)(10,12)(14,16,19,25)(15,20,23,36)(17,29,32,45)(18,34,30,27)(21,39,40,44)(22,41,33,37)(24,43,42,28)(26,38,35,31);(7,11)(10,12)

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, 26749);