C₂³×C₅×SD₃₂ #11985346

order	1280
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂³×C₅×SD₃₂

	subgroup	G/H
center, socle	C₂⁴×C₅	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₅
prime radical	C₂³×C₅×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

2334 conjugacy classes of subgroups; only the 1054 normal subgroups are shown.

normal subgroups only

This group has 2334 conjugacy classes of subgroups — too many to draw. Only its 1054 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 5 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 55614);