C₅×#33564 #11956284

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#33564

Subgroups

	subgroup	G/H
center	C₄×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₅
socle	C₂×C₅	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

1112 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 26552);