C₅×#12772 #11935492

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12772

Subgroups

	subgroup	G/H
center	C₄×C₅	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
socle	C₂×C₅	#1790
prime radical	C₅×#3529	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

760 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 37 points:

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

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, 5760);