C₅×#12862 #11935582

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12862

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁴×C₅ ◁ C₂³×C₅×Q₈ ◁ G

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

normal subgroups only

This group has 566 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 5850);