#65535

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #456

Subgroups

	subgroup	G/H
center	C₂	#4099
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting, prime radical	C₅×2₋¹⁺⁴	C₂
socle	C₂×C₅	#120
hypercenter	2₋¹⁺⁴	D₅
hypocenter	C₅	#456
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

2^6,5|a^2=f,c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^4,d^b=df,e^b=ef,e^c=ef

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598836, #10601781, #10601782, …
maximal normal subgroup of	#10598836, #10601781, #10601782, …
maximal quotient of	#10602451, #10602453, #10602457, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 873);