C₅×#7605 #11930325

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#7605

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

440 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 1280

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 593);