C₂×C₅×#2840 #11959391

order	1280
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

C₂×C₅×#2840

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂⁶×C₅	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

3550 conjugacy classes of subgroups; only the 974 normal subgroups are shown.

normal subgroups only

This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 974 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) = 31
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 5 generators on 31 points:

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

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, 29659);