C₅×#12971 #11935691

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12971

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×Q₈ ◁ G

424 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

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

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, 5959);